ordinal7.miz



    begin

    theorem :: ORDINAL7:1

    

     Th1: for X be set holds (X /\ ( succ X)) = X

    proof

      let X be set;

      for x be object holds x in X & x in ( succ X) iff x in X by XBOOLE_0:def 3;

      hence thesis by XBOOLE_0:def 4;

    end;

    registration

      let A be increasing Ordinal-Sequence, a be Ordinal;

      cluster (A | a) -> increasing;

      coherence

      proof

        now

          let c,d be Ordinal;

          assume

           A1: c in d & d in ( dom (A | a));

          then

           A2: ((A | a) . d) = (A . d) & ((A | a) . c) = (A . c) by FUNCT_1: 47, ORDINAL1: 10;

          d in ( dom A) by A1, RELAT_1: 57;

          hence ((A | a) . c) in ((A | a) . d) by A1, A2, ORDINAL2:def 12;

        end;

        hence thesis by ORDINAL2:def 12;

      end;

    end

    

     Lm1: ( succ 0 ) = 1;

    

     Lm2: ( succ ( succ 0 )) = 2;

    theorem :: ORDINAL7:2

    

     Th2: for a be Ordinal holds (a +^ a) = (2 *^ a)

    proof

      let a be Ordinal;

      consider fi be Ordinal-Sequence such that

       A1: (2 *^ a) = ( last fi) & ( dom fi) = ( succ 2) & (fi . 0 ) = 0 and

       A2: for c be Ordinal st ( succ c) in ( succ 2) holds (fi . ( succ c)) = ((fi . c) +^ a) and for c be Ordinal st c in ( succ 2) & c <> 0 & c is limit_ordinal holds (fi . c) = ( union ( sup (fi | c))) by ORDINAL2:def 15;

      ( succ 0 ) in ( succ ( succ 0 )) & ( succ ( succ 0 )) in ( succ 2) by ORDINAL1: 6;

      then

       A3: ( succ 0 ) in ( succ 2) & ( succ ( succ 0 )) in ( succ 2) by ORDINAL1: 10;

      (2 *^ a) = (fi . 2) by A1, ORDINAL2: 6

      .= ((fi . ( succ 0 )) +^ a) by A2, A3

      .= (((fi . 0 ) +^ a) +^ a) by A2, A3

      .= (a +^ a) by A1, ORDINAL2: 30;

      hence thesis;

    end;

    theorem :: ORDINAL7:3

    for a,b be Ordinal st 1 in a & a in b holds (b +^ a) in (a *^ b)

    proof

      let a,b be Ordinal;

      assume

       A1: 1 in a & a in b;

      then

       A2: (2 *^ b) c= (a *^ b) by Lm2, ORDINAL1: 21, ORDINAL2: 41;

      (b +^ a) in (b +^ b) by A1, ORDINAL2: 32;

      then (b +^ a) in (2 *^ b) by Th2;

      hence thesis by A2;

    end;

    theorem :: ORDINAL7:4

    

     Th4: for a be Ordinal holds (a *^ a) = ( exp (a,2))

    proof

      let a be Ordinal;

      consider fi be Ordinal-Sequence such that

       A1: ( exp (a,2)) = ( last fi) & ( dom fi) = ( succ 2) & (fi . 0 ) = 1 and

       A2: for c be Ordinal st ( succ c) in ( succ 2) holds (fi . ( succ c)) = (a *^ (fi . c)) and for c be Ordinal st c in ( succ 2) & c <> 0 & c is limit_ordinal holds (fi . c) = ( lim (fi | c)) by ORDINAL2:def 16;

      ( succ 0 ) in ( succ ( succ 0 )) & ( succ ( succ 0 )) in ( succ 2) by ORDINAL1: 6;

      then

       A3: ( succ 0 ) in ( succ 2) & ( succ ( succ 0 )) in ( succ 2) by ORDINAL1: 10;

      ( exp (a,2)) = (fi . 2) by A1, ORDINAL2: 6

      .= (a *^ (fi . ( succ 0 ))) by A2, A3

      .= (a *^ (a *^ (fi . 0 ))) by A2, A3

      .= (a *^ a) by A1, ORDINAL2: 39;

      hence thesis;

    end;

    theorem :: ORDINAL7:5

    for a,b be Ordinal st 1 in a & a in b holds (a *^ b) in ( exp (b,a))

    proof

      let a,b be Ordinal;

      assume

       A1: 1 in a & a in b;

      then

       A3: ( exp (b,2)) c= ( exp (b,a)) by Lm2, ORDINAL1: 21, ORDINAL4: 27;

      (a *^ b) in (b *^ b) by A1, ORDINAL2: 40;

      then (a *^ b) in ( exp (b,2)) by Th4;

      hence thesis by A3;

    end;

    theorem :: ORDINAL7:6

    for a,b be Ordinal st 1 in a & a in b holds ( exp (b,a)) in (b |^|^ a)

    proof

      let a,b be Ordinal;

      assume

       A1: 1 in a & a in b;

      then

       A2: 1 in b by ORDINAL1: 10;

      then 0 c< b by XBOOLE_1: 2, XBOOLE_0:def 8;

      then 0 in b by ORDINAL1: 11;

      then

       A3: (b |^|^ 2) c= (b |^|^ a) by A1, Lm2, ORDINAL1: 21, ORDINAL5: 21;

      ( exp (b,a)) in ( exp (b,b)) by A1, A2, ORDINAL4: 24;

      then ( exp (b,a)) in (b |^|^ 2) by ORDINAL5: 18;

      hence thesis by A3;

    end;

    registration

      cluster infinite for Ordinal-Sequence;

      existence

      proof

        take ( omega --> omega );

        ( dom ( omega --> omega )) = omega ;

        hence thesis;

      end;

    end

    

     Th9: for A,B be Sequence holds ( rng (A ^ B)) = (( rng A) \/ ( rng B)) by ORDINAL4: 2;

    theorem :: ORDINAL7:7

    

     Th10: for A,B be Sequence st (A ^ B) is Ordinal-yielding holds A is Ordinal-yielding & B is Ordinal-yielding

    proof

      let A,B be Sequence;

      assume (A ^ B) is Ordinal-yielding;

      then

      consider c be Ordinal such that

       A1: ( rng (A ^ B)) c= c by ORDINAL2:def 4;

      ( rng A) c= ( rng (A ^ B)) by ORDINAL4: 39;

      hence A is Ordinal-yielding by A1, XBOOLE_1: 1, ORDINAL2:def 4;

      ( rng B) c= ( rng (A ^ B)) by ORDINAL4: 40;

      hence B is Ordinal-yielding by A1, XBOOLE_1: 1, ORDINAL2:def 4;

    end;

    

     Th13: for D be set, p be FinSequence of D, n be Nat holds (n + 1) in ( dom p) iff n in ( dom ( FS2XFS p)) by AFINSQ_1: 94;

    

     Th15: for D be set, p be FinSequence of D holds ( rng p) = ( rng ( FS2XFS p)) by AFINSQ_1: 96;

    

     Th19: for D be set, p be one-to-one XFinSequence of D, n be Nat holds ( rng (p | n)) misses ( rng (p /^ n)) by AFINSQ_2: 87;

    theorem :: ORDINAL7:8

    

     Th21: for a,b be Ordinal st a in b holds (b -exponent a) = 0

    proof

      let a,b be Ordinal;

      assume

       A1: a in b;

      per cases ;

        suppose 0 in a;

        

        then 0 = (b -exponent (a *^ ( exp (b, 0 )))) by A1, ORDINAL5: 58

        .= (b -exponent (a *^ 1)) by ORDINAL2: 43

        .= (b -exponent a) by ORDINAL2: 39;

        hence thesis;

      end;

        suppose not 0 in a;

        hence thesis by ORDINAL5:def 10;

      end;

    end;

    theorem :: ORDINAL7:9

    

     Th22: for a,b,c be Ordinal st a c= c holds (b -exponent a) c= (b -exponent c)

    proof

      let a,b,c be Ordinal;

      assume

       A1: a c= c;

      per cases ;

        suppose

         A2: 1 in b & 0 in a & 0 in c;

        then ( exp (b,(b -exponent a))) c= a by ORDINAL5:def 10;

        then ( exp (b,(b -exponent a))) c= c by A1, XBOOLE_1: 1;

        hence thesis by A2, ORDINAL5:def 10;

      end;

        suppose not 1 in b;

        then (b -exponent a) = 0 & (b -exponent c) = 0 by ORDINAL5:def 10;

        hence thesis;

      end;

        suppose not 0 in a or not 0 in c;

        then not 0 in a by A1;

        then (b -exponent a) = {} by ORDINAL5:def 10;

        hence thesis;

      end;

    end;

    theorem :: ORDINAL7:10

    

     Th23: for a,b,c be Ordinal st 0 in a & 1 in b & a in ( exp (b,c)) holds (b -exponent a) in c

    proof

      let a,b,c be Ordinal;

      assume that

       A1: 0 in a and

       A2: 1 in b and

       A3: a in ( exp (b,c));

      ( exp (b,c)) = (1 *^ ( exp (b,c))) & 0 in 1 by CARD_1: 49, TARSKI:def 1, ORDINAL2: 39;

      then (b -exponent ( exp (b,c))) = c by A2, ORDINAL5: 58;

      then

       A4: (b -exponent a) c= c by A3, Th22, ORDINAL1:def 2;

      (b -exponent a) <> c

      proof

        assume

         A5: (b -exponent a) = c;

        ( exp (b,(b -exponent a))) c= a by A2, A1, ORDINAL5:def 10;

        hence contradiction by A3, A5, ORDINAL1: 5;

      end;

      hence thesis by A4, XBOOLE_0:def 8, ORDINAL1: 11;

    end;

    registration

      cluster decreasing -> one-to-one for Ordinal-Sequence;

      coherence

      proof

        let A be Ordinal-Sequence;

        assume

         A1: A is decreasing;

        now

          let x1,x2 be object;

          assume

           A2: x1 in ( dom A) & x2 in ( dom A) & (A . x1) = (A . x2);

          then

          reconsider a1 = x1, a2 = x2 as Ordinal;

          per cases by ORDINAL1: 14;

            suppose a1 in a2;

            then (A . a2) in (A . a1) by A1, A2, ORDINAL5:def 1;

            hence x1 = x2 by A2;

          end;

            suppose a1 = a2;

            hence x1 = x2;

          end;

            suppose a2 in a1;

            then (A . a1) in (A . a2) by A1, A2, ORDINAL5:def 1;

            hence x1 = x2 by A2;

          end;

        end;

        hence thesis by FUNCT_1:def 4;

      end;

    end

    registration

      let A be decreasing Sequence, a be Ordinal;

      cluster (A | a) -> decreasing;

      coherence

      proof

        now

          let b,c be Ordinal;

          assume

           A1: b in c & c in ( dom (A | a));

          then

           A2: ((A | a) . b) = (A . b) & ((A | a) . c) = (A . c) by FUNCT_1: 47, ORDINAL1: 10;

          c in ( dom A) by A1, RELAT_1: 57;

          hence ((A | a) . c) in ((A | a) . b) by A1, A2, ORDINAL5:def 1;

        end;

        hence thesis by ORDINAL5:def 1;

      end;

    end

    registration

      let A be non-decreasing Sequence, a be Ordinal;

      cluster (A | a) -> non-decreasing;

      coherence

      proof

        now

          let b,c be Ordinal;

          assume

           A1: b in c & c in ( dom (A | a));

          then

           A2: ((A | a) . b) = (A . b) & ((A | a) . c) = (A . c) by ORDINAL1: 10, FUNCT_1: 47;

          c in ( dom A) by A1, RELAT_1: 57;

          hence ((A | a) . b) c= ((A | a) . c) by A1, A2, ORDINAL5:def 2;

        end;

        hence thesis by ORDINAL5:def 2;

      end;

    end

    registration

      let A be non-increasing Sequence, a be Ordinal;

      cluster (A | a) -> non-increasing;

      coherence

      proof

        now

          let b,c be Ordinal;

          assume

           A1: b in c & c in ( dom (A | a));

          then

           A2: ((A | a) . b) = (A . b) & ((A | a) . c) = (A . c) by ORDINAL1: 10, FUNCT_1: 47;

          c in ( dom A) by A1, RELAT_1: 57;

          hence ((A | a) . c) c= ((A | a) . b) by A1, A2, ORDINAL5:def 3;

        end;

        hence thesis by ORDINAL5:def 3;

      end;

    end

    theorem :: ORDINAL7:11

    

     Th24: for A,B be finite Ordinal-Sequence holds ( Sum^ (A ^ B)) = (( Sum^ A) +^ ( Sum^ B))

    proof

      defpred P[ Nat] means for A,B be finite Ordinal-Sequence st ( dom B) = $1 holds ( Sum^ (A ^ B)) = (( Sum^ A) +^ ( Sum^ B));

      

       A1: P[ 0 ]

      proof

        let A,B be finite Ordinal-Sequence;

        assume ( dom B) = 0 ;

        then B = {} ;

        hence ( Sum^ (A ^ B)) = (( Sum^ A) +^ ( Sum^ B)) by ORDINAL2: 27, ORDINAL5: 52;

      end;

      

       A2: for n be Nat st P[n] holds P[(n + 1)]

      proof

        let n be Nat;

        assume

         A3: P[n];

        let A,B be finite Ordinal-Sequence;

        assume

         A4: ( dom B) = (n + 1);

        then B <> {} ;

        then

        consider C be XFinSequence, a be object such that

         A5: B = (C ^ <%a%>) by AFINSQ_1: 40;

        consider b be Ordinal such that

         A6: ( rng B) c= b by ORDINAL2:def 4;

        ( rng C) c= ( rng B) by A5, AFINSQ_1: 24;

        then

        reconsider C as finite Ordinal-Sequence by A6, XBOOLE_1: 1, ORDINAL2:def 4;

        ( rng <%a%>) c= ( rng B) by A5, AFINSQ_1: 25;

        then {a} c= ( rng B) by AFINSQ_1: 33;

        then a in ( rng B) by ZFMISC_1: 31;

        then

        reconsider a as Ordinal;

        

         A7: (( dom C) + 1) = (( len C) + ( len <%a%>)) by AFINSQ_1: 34

        .= (n + 1) by A4, A5, AFINSQ_1: 17;

        

        thus ( Sum^ (A ^ B)) = ( Sum^ ((A ^ C) ^ <%a%>)) by A5, AFINSQ_1: 27

        .= (( Sum^ (A ^ C)) +^ a) by ORDINAL5: 54

        .= ((( Sum^ A) +^ ( Sum^ C)) +^ a) by A3, A7

        .= (( Sum^ A) +^ (( Sum^ C) +^ a)) by ORDINAL3: 30

        .= (( Sum^ A) +^ ( Sum^ B)) by A5, ORDINAL5: 54;

      end;

      

       A8: for n be Nat holds P[n] from NAT_1:sch 2( A1, A2);

      let A,B be finite Ordinal-Sequence;

      ( dom B) is Nat;

      hence thesis by A8;

    end;

    theorem :: ORDINAL7:12

    

     Th25: for a,b be Ordinal holds ( Sum^ <%a, b%>) = (a +^ b)

    proof

      let a,b be Ordinal;

      

      thus ( Sum^ <%a, b%>) = ( Sum^ ( <%a%> ^ <%b%>)) by AFINSQ_1:def 5

      .= (( Sum^ <%a%>) +^ ( Sum^ <%b%>)) by Th24

      .= (( Sum^ <%a%>) +^ b) by ORDINAL5: 53

      .= (a +^ b) by ORDINAL5: 53;

    end;

    registration

      let A be non empty non-empty finite Ordinal-Sequence;

      cluster ( Sum^ A) -> non empty;

      coherence

      proof

         0 c< ( dom A) by XBOOLE_1: 2, XBOOLE_0:def 8;

        then

         A1: 0 in ( dom A) by ORDINAL1: 11;

        (A . 0 ) c= ( Sum^ A) by ORDINAL5: 56;

        hence thesis by A1;

      end;

      let B be finite Ordinal-Sequence;

      cluster ( Sum^ (A ^ B)) -> non empty;

      coherence

      proof

        ( Sum^ (A ^ B)) = (( Sum^ A) +^ ( Sum^ B)) by Th24;

        hence thesis;

      end;

      cluster ( Sum^ (B ^ A)) -> non empty;

      coherence

      proof

        ( Sum^ (B ^ A)) = (( Sum^ B) +^ ( Sum^ A)) by Th24;

        hence thesis;

      end;

    end

    theorem :: ORDINAL7:13

    

     Th26: for a be Ordinal, n be Nat holds ( Sum^ (n --> a)) = (n *^ a)

    proof

      let a be Ordinal, n be Nat;

      consider fi be Ordinal-Sequence such that

       A1: ( Sum^ (n --> a)) = ( last fi) & ( dom fi) = ( succ ( dom (n --> a))) & (fi . 0 ) = 0 and

       A2: for k be Nat st k in ( dom (n --> a)) holds (fi . (k + 1)) = ((fi . k) +^ ((n --> a) . k)) by ORDINAL5:def 8;

       A4:

      now

        let C be Ordinal;

        assume ( succ C) in ( succ n);

        then

         A5: C in n by ORDINAL3: 3;

        n in omega by ORDINAL1:def 12;

        then C in omega by A5, ORDINAL1: 10;

        then

        reconsider k = C as Nat;

        

         A6: k in ( dom (n --> a)) by A5;

        

        thus (fi . ( succ C)) = (fi . ( succ ( Segm C)))

        .= (fi . ( Segm (k + 1))) by NAT_1: 38

        .= ((fi . k) +^ ((n --> a) . k)) by A2, A6

        .= ((fi . C) +^ a) by A5, FUNCOP_1: 7;

      end;

      now

        let C be Ordinal;

        assume

         A7: C in ( succ n) & C <> 0 & C is limit_ordinal;

        ( succ n) in omega by ORDINAL1:def 12;

        then C in omega by A7, ORDINAL1: 10;

        hence (fi . C) = ( union ( sup (fi | C))) by A7;

      end;

      hence thesis by A1, A4, ORDINAL2:def 15;

    end;

    

     Lm5: for n be Nat holds ( succ n) = (n + 1)

    proof

      let n be Nat;

      

      thus ( succ n) = ( succ ( Segm n))

      .= ( Segm (n + 1)) by NAT_1: 38

      .= (n + 1);

    end;

    theorem :: ORDINAL7:14

    

     Th27: for A be finite Ordinal-Sequence, a be Ordinal holds ( Sum^ (A | a)) c= ( Sum^ A)

    proof

      let A be finite Ordinal-Sequence, a be Ordinal;

      per cases ;

        suppose ( dom A) c= a;

        hence thesis by RELAT_1: 68;

      end;

        suppose

         A1: a c= ( dom A);

        then

        reconsider a as finite Ordinal;

        consider f1 be Ordinal-Sequence such that

         A2: ( Sum^ (A | a)) = ( last f1) & ( dom f1) = ( succ ( dom (A | a))) & (f1 . 0 ) = 0 and

         A3: for n be Nat st n in ( dom (A | a)) holds (f1 . (n + 1)) = ((f1 . n) +^ ((A | a) . n)) by ORDINAL5:def 8;

        consider f2 be Ordinal-Sequence such that

         A4: ( Sum^ A) = ( last f2) & ( dom f2) = ( succ ( dom A)) & (f2 . 0 ) = 0 and

         A5: for n be Nat st n in ( dom A) holds (f2 . (n + 1)) = ((f2 . n) +^ (A . n)) by ORDINAL5:def 8;

        defpred P[ Nat] means $1 in ( dom f1) implies (f1 . $1) = (f2 . $1);

        

         A6: P[ 0 ] by A2, A4;

        

         A7: for n be Nat st P[n] holds P[(n + 1)]

        proof

          let n be Nat;

          assume

           A8: P[n];

          assume (n + 1) in ( dom f1);

          then

           A9: ( succ n) in ( dom f1) by Lm5;

          n in ( succ n) by ORDINAL1: 6;

          then

           A10: (f1 . n) = (f2 . n) by A8, A9, ORDINAL1: 10;

          

           A11: n in ( dom (A | a)) by A2, A9, ORDINAL3: 3;

          then

           A12: n in ( dom A) by RELAT_1: 57;

          

          thus (f1 . (n + 1)) = ((f1 . n) +^ ((A | a) . n)) by A3, A11

          .= ((f2 . n) +^ (A . n)) by A10, A11, FUNCT_1: 47

          .= (f2 . (n + 1)) by A5, A12;

        end;

        

         A13: for n be Nat holds P[n] from NAT_1:sch 2( A6, A7);

        

         A14: ( last f1) = (f1 . ( dom (A | a))) & ( last f2) = (f2 . ( dom A)) by A2, A4, ORDINAL2: 6;

        

        then

         A15: ( last f1) = (f2 . ( dom (A | a))) by A2, A13, ORDINAL1: 6

        .= (f2 . a) by A1, RELAT_1: 62;

        ( Segm a) c= ( Segm ( dom A)) by A1;

        then

        consider k be Nat such that

         A16: ( dom A) = (a + k) by NAT_1: 10, NAT_1: 39;

        defpred Q[ Nat] means (a + $1) <= ( dom A) implies (f2 . a) c= (f2 . (a + $1));

        

         A17: Q[ 0 ];

        

         A18: for n be Nat st Q[n] holds Q[(n + 1)]

        proof

          let n be Nat;

          assume

           A19: Q[n];

          assume

           A20: (a + (n + 1)) <= ( dom A);

          then ((a + n) + 1) < (( dom A) + 1) by NAT_1: 13;

          then

           A21: (f2 . a) c= (f2 . (a + n)) by A19, XREAL_1: 6;

          ( Segm (a + (n + 1))) c= ( Segm ( dom A)) by A20, NAT_1: 39;

          then ((a + n) + 1) c= ( dom A);

          then ( succ (a + n)) c= ( dom A) by Lm5;

          then (f2 . ((a + n) + 1)) = ((f2 . (a + n)) +^ (A . (a + n))) by A5, ORDINAL1: 21;

          then (f2 . (a + n)) c= (f2 . ((a + n) + 1)) by ORDINAL3: 24;

          hence thesis by A21, XBOOLE_1: 1;

        end;

        for n be Nat holds Q[n] from NAT_1:sch 2( A17, A18);

        hence thesis by A2, A4, A14, A15, A16;

      end;

    end;

    theorem :: ORDINAL7:15

    

     Th28: for A,B be finite Ordinal-Sequence st ( dom A) c= ( dom B) & for a be object st a in ( dom A) holds (A . a) c= (B . a) holds ( Sum^ A) c= ( Sum^ B)

    proof

      let A,B be finite Ordinal-Sequence;

      assume that

       A1: ( dom A) c= ( dom B) and

       A2: for a be object st a in ( dom A) holds (A . a) c= (B . a);

      set a = ( dom A);

      consider f1 be Ordinal-Sequence such that

       A3: ( Sum^ A) = ( last f1) & ( dom f1) = ( succ ( dom A)) & (f1 . 0 ) = 0 and

       A4: for n be Nat st n in ( dom A) holds (f1 . (n + 1)) = ((f1 . n) +^ (A . n)) by ORDINAL5:def 8;

      consider f2 be Ordinal-Sequence such that

       A5: ( Sum^ (B | a)) = ( last f2) & ( dom f2) = ( succ ( dom (B | a))) & (f2 . 0 ) = 0 and

       A6: for n be Nat st n in ( dom (B | a)) holds (f2 . (n + 1)) = ((f2 . n) +^ ((B | a) . n)) by ORDINAL5:def 8;

      defpred P[ Nat] means $1 in ( succ a) implies (f1 . $1) c= (f2 . $1);

      

       A7: P[ 0 ] by A3;

      

       A8: for n be Nat st P[n] holds P[(n + 1)]

      proof

        let n be Nat;

        assume

         A9: P[n];

        assume (n + 1) in ( succ a);

        then

         A10: ( succ n) in ( succ a) by Lm5;

        then

         A11: n in a by ORDINAL3: 3;

        n in ( succ n) by ORDINAL1: 6;

        then

         A12: (f1 . n) c= (f2 . n) by A9, A10, ORDINAL1: 10;

        

         A13: (f1 . (n + 1)) = ((f1 . n) +^ (A . n)) by A4, A10, ORDINAL3: 3;

        

         A14: n in ( dom (B | a)) by A1, A11, RELAT_1: 62;

        

        then

         A15: (f2 . (n + 1)) = ((f2 . n) +^ ((B | a) . n)) by A6

        .= ((f2 . n) +^ (B . n)) by A14, FUNCT_1: 47;

        (A . n) c= (B . n) by A2, A10, ORDINAL3: 3;

        hence thesis by A12, A13, A15, ORDINAL3: 18;

      end;

      for n be Nat holds P[n] from NAT_1:sch 2( A7, A8);

      then (f1 . a) c= (f2 . a) by ORDINAL1: 6;

      then ( last f1) c= (f2 . a) by A3, ORDINAL2: 6;

      then ( last f1) c= (f2 . ( dom (B | a))) by A1, RELAT_1: 62;

      then

       A17: ( Sum^ A) c= ( Sum^ (B | a)) by A3, A5, ORDINAL2: 6;

      ( Sum^ (B | a)) c= ( Sum^ B) by Th27;

      hence thesis by A17, XBOOLE_1: 1;

    end;

    theorem :: ORDINAL7:16

    

     Th29: for A be Cantor-normal-form Ordinal-Sequence st A <> {} holds ex B be Cantor-normal-form Ordinal-Sequence, a be Cantor-component Ordinal st A = (B ^ <%a%>)

    proof

      let A be Cantor-normal-form Ordinal-Sequence;

      assume A <> {} ;

      then

      consider B be XFinSequence, a be object such that

       A1: A = (B ^ <%a%>) by AFINSQ_1: 40;

      reconsider B as finite Ordinal-Sequence by A1, Th10;

       <%a%> is Ordinal-Sequence by A1, Th10;

      then

      consider c be Ordinal such that

       A2: ( rng <%a%>) c= c by ORDINAL2:def 4;

       {a} c= c by A2, AFINSQ_1: 33;

      then a in c by ZFMISC_1: 31;

      then

      reconsider a as Ordinal;

      ( len A) = (( len B) + ( len <%a%>)) by A1, AFINSQ_1: 17

      .= ( Segm (( len B) + 1)) by AFINSQ_1: 34

      .= ( succ ( Segm ( len B))) by NAT_1: 38

      .= ( succ ( len B));

      then ( len B) in ( len A) by ORDINAL1: 6;

      then (A . ( len B)) is Cantor-component by ORDINAL5:def 11;

      then

      reconsider a as Cantor-component Ordinal by A1, AFINSQ_1: 36;

      ( dom B) c= (( dom B) +^ ( dom <%a%>)) by ORDINAL3: 24;

      then

       A3: ( dom B) c= ( dom A) by A1, ORDINAL4:def 1;

       A4:

      now

        let b be Ordinal;

        assume

         A5: b in ( dom B);

        then (A . b) = (B . b) by A1, ORDINAL4:def 1;

        hence (B . b) is Cantor-component by A3, A5, ORDINAL5:def 11;

      end;

      now

        let b,c be Ordinal;

        assume

         A6: b in c & c in ( dom B);

        then b in ( dom B) & c in ( dom B) by ORDINAL1: 10;

        then (A . b) = (B . b) & (A . c) = (B . c) by A1, ORDINAL4:def 1;

        hence ( omega -exponent (B . c)) in ( omega -exponent (B . b)) by A3, A6, ORDINAL5:def 11;

      end;

      then

      reconsider B as Cantor-normal-form Ordinal-Sequence by A4, ORDINAL5:def 11;

      take B, a;

      thus thesis by A1;

    end;

    registration

      let A be Cantor-normal-form Ordinal-Sequence, n be Nat;

      cluster (A | n) -> Cantor-normal-form;

      coherence

      proof

         A1:

        now

          let a be Ordinal;

          assume a in ( dom (A | n));

          then a in ( dom A) & ((A | n) . a) = (A . a) by RELAT_1: 57, FUNCT_1: 47;

          hence ((A | n) . a) is Cantor-component by ORDINAL5:def 11;

        end;

        now

          let a,b be Ordinal;

          assume

           A2: a in b & b in ( dom (A | n));

          then

           A3: a in ( dom (A | n)) & b in ( dom A) by ORDINAL1: 10, RELAT_1: 57;

          then ((A | n) . a) = (A . a) & ((A | n) . b) = (A . b) by A2, FUNCT_1: 47;

          hence ( omega -exponent ((A | n) . b)) in ( omega -exponent ((A | n) . a)) by A2, A3, ORDINAL5:def 11;

        end;

        hence thesis by A1, ORDINAL5:def 11;

      end;

    end

    registration

      let A be Cantor-normal-form Ordinal-Sequence, n be Nat;

      cluster (A /^ n) -> Cantor-normal-form;

      coherence

      proof

        per cases ;

          suppose n >= ( len A);

          hence thesis by AFINSQ_2: 6;

        end;

          suppose

           A1: n < ( len A);

           A2:

          now

            let a be Ordinal;

            assume a in ( dom (A /^ n));

            then ((A /^ n) . a) in ( rng (A /^ n)) by FUNCT_1: 3;

            then ((A /^ n) . a) in ( rng A) by AFINSQ_2: 9, TARSKI:def 3;

            then

            consider b be object such that

             A3: b in ( dom A) & (A . b) = ((A /^ n) . a) by FUNCT_1:def 3;

            thus ((A /^ n) . a) is Cantor-component by A3, ORDINAL5:def 11;

          end;

          now

            let a,b be Ordinal;

            assume

             A4: a in b & b in ( dom (A /^ n));

            then

             A5: a in ( dom (A /^ n)) by ORDINAL1: 10;

            then

            reconsider m = a, k = b as Nat by A4;

            ((A /^ n) . a) = (A . (m + n)) & ((A /^ n) . b) = (A . (k + n)) by A4, A5, AFINSQ_2:def 2;

            then

             A6: ((A /^ n) . a) = (A . (a +^ n)) & ((A /^ n) . b) = (A . (b +^ n)) by CARD_2: 36;

            

             A7: ((( dom A) - n) + n) = (( len (A /^ n)) + n) by A1, AFINSQ_2: 7

            .= (( len (A /^ n)) +^ n) by CARD_2: 36;

            m in ( Segm k) by A4;

            then (m + n) < (k + n) by NAT_1: 44, XREAL_1: 6;

            then (m + n) in ( Segm (k + n)) by NAT_1: 44;

            then (m + n) in (b +^ n) by CARD_2: 36;

            then

             A8: (a +^ n) in (b +^ n) by CARD_2: 36;

            k in ( Segm ( dom (A /^ n))) by A4;

            then (k + n) < (( dom (A /^ n)) + n) by NAT_1: 44, XREAL_1: 6;

            then (k + n) in ( Segm (( dom (A /^ n)) + n)) by NAT_1: 44;

            then (b +^ n) in (( dom (A /^ n)) + n) by CARD_2: 36;

            then (b +^ n) in ( dom A) by A7, CARD_2: 36;

            hence ( omega -exponent ((A /^ n) . b)) in ( omega -exponent ((A /^ n) . a)) by A6, A8, ORDINAL5:def 11;

          end;

          hence thesis by A2, ORDINAL5:def 11;

        end;

      end;

    end

    registration

      cluster natural-valued -> Ordinal-yielding for Sequence;

      coherence

      proof

        let F be Sequence;

        assume F is natural-valued;

        then ( rng F) c= NAT by VALUED_0:def 6;

        hence thesis by ORDINAL2:def 4;

      end;

    end

    registration

      cluster limit_ordinal -> zero for Nat;

      coherence ;

      cluster non limit_ordinal for Ordinal;

      existence

      proof

        take the non zero Nat;

        thus thesis;

      end;

    end

    registration

      let n,m be Nat;

      identify max (n,m) with n \/ m;

      correctness

      proof

        per cases by ORDINAL1: 14;

          suppose

           A1: n in m;

          then

           A2: (n \/ m) = m by ORDINAL1:def 2, XBOOLE_1: 12;

          n in ( Segm m) by A1;

          hence thesis by A2, NAT_1: 44, XXREAL_0:def 10;

        end;

          suppose n = m;

          hence thesis;

        end;

          suppose

           A3: m in n;

          then

           A4: (n \/ m) = n by ORDINAL1:def 2, XBOOLE_1: 12;

          m in ( Segm n) by A3;

          hence thesis by A4, NAT_1: 44, XXREAL_0:def 10;

        end;

      end;

      identify min (n,m) with n /\ m;

      correctness

      proof

        per cases by ORDINAL1: 14;

          suppose

           A5: n in m;

          then

           A6: (n /\ m) = n by ORDINAL1:def 2, XBOOLE_1: 28;

          n in ( Segm m) by A5;

          hence thesis by A6, NAT_1: 44, XXREAL_0:def 9;

        end;

          suppose n = m;

          hence thesis;

        end;

          suppose

           A7: m in n;

          then

           A8: (n /\ m) = m by ORDINAL1:def 2, XBOOLE_1: 28;

          m in ( Segm n) by A7;

          hence thesis by A8, NAT_1: 44, XXREAL_0:def 9;

        end;

      end;

    end

    begin

    theorem :: ORDINAL7:17

    

     Th30: for a,b be Ordinal holds (a +^ b) = b iff ( omega *^ a) c= b

    proof

      let a,b be Ordinal;

      hereby

        assume

         A1: (a +^ b) = b;

        defpred P[ Nat] means (($1 *^ a) +^ b) = b;

        ( 0 *^ a) = 0 by ORDINAL2: 35;

        then

         A2: P[ 0 ] by ORDINAL2: 30;

        

         A3: for n be Nat st P[n] holds P[(n + 1)]

        proof

          let n be Nat;

          assume

           A4: P[n];

          

          thus (((n + 1) *^ a) +^ b) = ((( succ n) *^ a) +^ b) by Lm5

          .= (((n *^ a) +^ a) +^ b) by ORDINAL2: 36

          .= b by A1, A4, ORDINAL3: 30;

        end;

        

         A5: for n be Nat holds P[n] from NAT_1:sch 2( A2, A3);

        per cases ;

          suppose a = {} ;

          then ( omega *^ a) = {} by ORDINAL2: 38;

          hence ( omega *^ a) c= b;

        end;

          suppose

           A6: a <> {} ;

          reconsider fi = ( id omega ) as Ordinal-Sequence;

          

           A7: ( sup fi) = ( sup ( rng fi)) by ORDINAL2:def 5

          .= omega by ORDINAL2: 18;

          set psi = (fi *^ a);

          

           A8: ( dom fi) = ( dom psi) by ORDINAL3:def 4;

          for A,B be Ordinal st A in ( dom fi) & B = (fi . A) holds (psi . A) = (B *^ a) by ORDINAL3:def 4;

          then

           A9: ( sup psi) = ( omega *^ a) by A6, A7, A8, ORDINAL3: 42;

          now

            let A be Ordinal;

            assume A in ( rng psi);

            then

            consider n be object such that

             A10: n in ( dom psi) & (psi . n) = A by FUNCT_1:def 3;

            reconsider n as Nat by A8, A10;

            A = ((fi . n) *^ a) by A8, A10, ORDINAL3:def 4

            .= (n *^ a) by A8, A10, FUNCT_1: 18;

            then

             A11: (A +^ b) = b by A5;

            then

             A12: A c= b by ORDINAL3: 24;

            A <> b

            proof

              assume A = b;

              

              then (2 *^ b) = (A +^ b) by Th2

              .= (1 *^ b) by A11, ORDINAL2: 39;

              hence contradiction by A1, A6, ORDINAL3: 33;

            end;

            hence A in b by A12, XBOOLE_0:def 8, ORDINAL1: 11;

          end;

          then ( sup ( rng psi)) c= b by ORDINAL2: 20;

          hence ( omega *^ a) c= b by A9, ORDINAL2:def 5;

        end;

      end;

      assume ( omega *^ a) c= b;

      then

      consider c be Ordinal such that

       A13: b = (( omega *^ a) +^ c) by ORDINAL3: 27;

      

      thus (a +^ b) = ((1 *^ a) +^ (( omega *^ a) +^ c)) by A13, ORDINAL2: 39

      .= (((1 *^ a) +^ ( omega *^ a)) +^ c) by ORDINAL3: 30

      .= (((1 +^ omega ) *^ a) +^ c) by ORDINAL3: 46

      .= b by A13, CARD_2: 74;

    end;

    theorem :: ORDINAL7:18

    

     Th31: for A be non empty Cantor-normal-form Ordinal-Sequence, a be object st a in ( dom A) holds ( omega -exponent ( last A)) c= ( omega -exponent (A . a))

    proof

      let A be non empty Cantor-normal-form Ordinal-Sequence, a be object;

      assume

       A1: a in ( dom A);

      consider A0 be Cantor-normal-form Ordinal-Sequence, a0 be Cantor-component Ordinal such that

       A2: A = (A0 ^ <%a0%>) by Th29;

      per cases by A1, A2, AFINSQ_1: 20;

        suppose

         A3: a in ( dom A0);

         0 in 1 by CARD_1: 49, TARSKI:def 1;

        then 0 in ( dom <%a0%>) by AFINSQ_1: 33;

        then (( len A0) + 0 ) in ( dom A) by A2, AFINSQ_1: 23;

        then ( omega -exponent (A . ( len A0))) in ( omega -exponent (A . a)) by A3, ORDINAL5:def 11;

        then ( omega -exponent a0) in ( omega -exponent (A . a)) by A2, AFINSQ_1: 36;

        then ( omega -exponent ( last A)) in ( omega -exponent (A . a)) by A2, AFINSQ_1: 92;

        hence thesis by ORDINAL1:def 2;

      end;

        suppose ex n be Nat st n in ( dom <%a0%>) & a = (( len A0) + n);

        then

        consider n be Nat such that

         A4: n in ( dom <%a0%>) & a = (( len A0) + n);

        n in ( Segm 1) by A4, AFINSQ_1: 33;

        then n = 0 by CARD_1: 49, TARSKI:def 1;

        

        then (A . a) = a0 by A2, A4, AFINSQ_1: 36

        .= ( last A) by A2, AFINSQ_1: 92;

        hence thesis;

      end;

    end;

    theorem :: ORDINAL7:19

    

     Th32: for A be non empty Cantor-normal-form Ordinal-Sequence, a be object st a in ( dom A) holds ( omega -exponent (A . a)) c= ( omega -exponent (A . 0 ))

    proof

      let A be non empty Cantor-normal-form Ordinal-Sequence, a be object;

      assume

       A1: a in ( dom A);

      consider a0 be Cantor-component Ordinal, A0 be Cantor-normal-form Ordinal-Sequence such that

       A2: A = ( <%a0%> ^ A0) by ORDINAL5: 67;

      per cases by A1, A2, AFINSQ_1: 20;

        suppose a in ( dom <%a0%>);

        then a in ( Segm 1) by AFINSQ_1: 33;

        hence thesis by CARD_1: 49, TARSKI:def 1;

      end;

        suppose ex n be Nat st n in ( dom A0) & a = (( len <%a0%>) + n);

        then

        consider n be Nat such that

         A3: n in ( dom A0) & a = (( len <%a0%>) + n);

        reconsider n1 = a as Nat by A3;

        n1 = (n + 1) by A3, AFINSQ_1: 34;

        then 0 in ( Segm n1) by NAT_1: 44;

        then

         A4: 0 in n1;

        n1 in ( dom A) by A2, A3, AFINSQ_1: 23;

        hence thesis by A4, ORDINAL5:def 11, ORDINAL1:def 2;

      end;

    end;

    theorem :: ORDINAL7:20

    

     Th33: for A,B be non empty Cantor-normal-form Ordinal-Sequence st ( omega -exponent (B . 0 )) in ( omega -exponent ( last A)) holds (A ^ B) is Cantor-normal-form

    proof

      let A,B be non empty Cantor-normal-form Ordinal-Sequence;

      assume

       A1: ( omega -exponent (B . 0 )) in ( omega -exponent ( last A));

       A2:

      now

        let a be Ordinal;

        assume a in ( dom (A ^ B));

        per cases by AFINSQ_1: 20;

          suppose

           A3: a in ( dom A);

          then (A . a) = ((A ^ B) . a) by AFINSQ_1:def 3;

          hence ((A ^ B) . a) is Cantor-component by A3, ORDINAL5:def 11;

        end;

          suppose ex n be Nat st n in ( dom B) & a = (( len A) + n);

          then

          consider n be Nat such that

           A4: n in ( dom B) & a = (( len A) + n);

          (B . n) = ((A ^ B) . a) by A4, AFINSQ_1:def 3;

          hence ((A ^ B) . a) is Cantor-component by A4, ORDINAL5:def 11;

        end;

      end;

      for a,b be Ordinal st a in b & b in ( dom (A ^ B)) holds ( omega -exponent ((A ^ B) . b)) in ( omega -exponent ((A ^ B) . a))

      proof

        let a,b be Ordinal;

        assume

         A5: a in b & b in ( dom (A ^ B));

        per cases by AFINSQ_1: 20;

          suppose

           A6: b in ( dom A);

          then

           A7: ((A ^ B) . b) = (A . b) & a in ( dom A) by A5, ORDINAL1: 10, AFINSQ_1:def 3;

          then ((A ^ B) . a) = (A . a) by AFINSQ_1:def 3;

          hence thesis by A5, A6, A7, ORDINAL5:def 11;

        end;

          suppose ex n be Nat st n in ( dom B) & b = (( len A) + n);

          then

          consider n be Nat such that

           A8: n in ( dom B) & b = (( len A) + n);

          a in ( dom (A ^ B)) by A5, ORDINAL1: 10;

          per cases by AFINSQ_1: 20;

            suppose

             A9: a in ( dom A);

            then ( omega -exponent ( last A)) c= ( omega -exponent (A . a)) by Th31;

            then

             A10: ( omega -exponent (B . 0 )) in ( omega -exponent (A . a)) by A1;

            ( omega -exponent (B . n)) c= ( omega -exponent (B . 0 )) by A8, Th32;

            then ( omega -exponent (B . n)) in ( omega -exponent (A . a)) by A10, ORDINAL1: 12;

            then ( omega -exponent ((A ^ B) . b)) in ( omega -exponent (A . a)) by A8, AFINSQ_1:def 3;

            hence thesis by A9, AFINSQ_1:def 3;

          end;

            suppose ex m be Nat st m in ( dom B) & a = (( len A) + m);

            then

            consider m be Nat such that

             A11: m in ( dom B) & a = (( len A) + m);

            m in n

            proof

              assume not m in n;

              then (( len A) +^ n) c= (( len A) +^ m) by ORDINAL1: 16, ORDINAL2: 33;

              then b c= (( len A) +^ m) by A8, CARD_2: 36;

              then b c= a by A11, CARD_2: 36;

              then a in a by A5;

              hence contradiction;

            end;

            then ( omega -exponent (B . n)) in ( omega -exponent (B . m)) by A8, ORDINAL5:def 11;

            then ( omega -exponent ((A ^ B) . b)) in ( omega -exponent (B . m)) by A8, AFINSQ_1:def 3;

            hence thesis by A11, AFINSQ_1:def 3;

          end;

        end;

      end;

      hence thesis by A2, ORDINAL5:def 11;

    end;

    

     Lm6: for A be decreasing Ordinal-Sequence, n be Nat st ( len A) = (n + 1) holds ( rng (A | n)) = (( rng A) \ {(A . n)})

    proof

      let A be decreasing Ordinal-Sequence, n be Nat;

      assume

       A1: ( len A) = (n + 1);

       not (A . n) in ( rng (A | n))

      proof

        assume (A . n) in ( rng (A | n));

        then

        consider x be object such that

         A2: x in ( dom (A | n)) & ((A | n) . x) = (A . n) by FUNCT_1:def 3;

        

         A3: (A . x) = (A . n) by A2, FUNCT_1: 47;

        

         A4: x in ( dom A) & x in n by A2, RELAT_1: 57;

        (n + 0 ) < (n + 1) by XREAL_1: 8;

        then n in ( dom A) by A1, AFINSQ_1: 86;

        then n in n by A3, A4, FUNCT_1:def 4;

        hence contradiction;

      end;

      then

       A5: ( rng (A | n)) c= (( rng A) \ {(A . n)}) by RELAT_1: 70, ZFMISC_1: 34;

      now

        let y be object;

        assume y in (( rng A) \ {(A . n)});

        then

         A6: y in ( rng A) & y <> (A . n) by ZFMISC_1: 56;

        then

        consider x be object such that

         A7: x in ( dom A) & (A . x) = y by FUNCT_1:def 3;

        ( dom A) = ( succ n) by A1, Lm5;

        then x in n by A6, A7, ORDINAL1: 8;

        hence y in ( rng (A | n)) by A7, FUNCT_1: 50;

      end;

      then (( rng A) \ {(A . n)}) c= ( rng (A | n)) by TARSKI:def 3;

      hence thesis by A5, XBOOLE_0:def 10;

    end;

    

     Lm7: for A,B be decreasing Ordinal-Sequence, n be Nat st ( len A) = (n + 1) & ( rng A) = ( rng B) holds (A . n) = (B . n)

    proof

      let A,B be decreasing Ordinal-Sequence, n be Nat;

      assume

       A1: ( len A) = (n + 1) & ( rng A) = ( rng B);

      

       A2: ( dom A) = ( card ( dom A))

      .= ( card ( rng B)) by A1, CARD_1: 70

      .= ( card ( dom B)) by CARD_1: 70

      .= ( dom B);

      n in ( succ n) by ORDINAL1: 6;

      then

       A3: n in (n + 1) by Lm5;

      then (A . n) in ( rng B) by A1, FUNCT_1: 3;

      then

      consider m be object such that

       A4: m in ( dom B) & (B . m) = (A . n) by FUNCT_1:def 3;

      (B . n) in ( rng A) by A1, A2, A3, FUNCT_1: 3;

      then

      consider k be object such that

       A5: k in ( dom A) & (A . k) = (B . n) by FUNCT_1:def 3;

      reconsider m, k as Nat by A4, A5;

      per cases by ORDINAL1: 14;

        suppose m in k;

        then

         A6: (A . k) in (A . m) & (B . k) in (B . m) by A2, A5, ORDINAL5:def 1;

        k in ( succ n) by A1, A5, Lm5;

        per cases by ORDINAL1: 8;

          suppose k in n;

          then (A . n) in (A . k) & (B . n) in (B . k) by A1, A2, A3, ORDINAL5:def 1;

          hence thesis by A4, A5, A6, ORDINAL1: 10;

        end;

          suppose k = n;

          hence thesis by A5;

        end;

      end;

        suppose

         A7: m = k;

        k in ( succ n) by A1, A5, Lm5;

        per cases by ORDINAL1: 8;

          suppose k in n;

          then (A . n) in (A . k) & (B . n) in (B . k) by A1, A2, A3, ORDINAL5:def 1;

          hence thesis by A4, A5, A7;

        end;

          suppose k = n;

          hence thesis by A4, A7;

        end;

      end;

        suppose k in m;

        then

         A8: (A . m) in (A . k) & (B . m) in (B . k) by A2, A4, ORDINAL5:def 1;

        m in ( succ n) by A1, A2, A4, Lm5;

        per cases by ORDINAL1: 8;

          suppose m in n;

          then (A . n) in (A . m) & (B . n) in (B . m) by A1, A2, A3, ORDINAL5:def 1;

          hence thesis by A4, A5, A8, ORDINAL1: 10;

        end;

          suppose m = n;

          hence thesis by A4;

        end;

      end;

    end;

    theorem :: ORDINAL7:21

    

     Th34: for A,B be decreasing Ordinal-Sequence st ( rng A) = ( rng B) holds A = B

    proof

      defpred P[ Nat] means for A,B be decreasing Ordinal-Sequence st ( len A) = $1 & ( rng A) = ( rng B) holds A = B;

      

       A1: P[ 0 ]

      proof

        let A,B be decreasing Ordinal-Sequence;

        assume

         A2: ( len A) = 0 & ( rng A) = ( rng B);

        then A is empty;

        hence thesis by A2;

      end;

      

       A3: for n be Nat st P[n] holds P[(n + 1)]

      proof

        let n be Nat;

        assume

         A4: P[n];

        let A,B be decreasing Ordinal-Sequence;

        assume

         A5: ( len A) = (n + 1) & ( rng A) = ( rng B);

        ( dom A) = ( card ( dom A))

        .= ( card ( rng B)) by A5, CARD_1: 70

        .= ( card ( dom B)) by CARD_1: 70

        .= ( dom B);

        then

         A6: ( len B) = (n + 1) by A5;

        set A0 = (A | n), B0 = (B | n);

        ( rng A0) = (( rng A) \ {(A . n)}) & ( rng B0) = (( rng B) \ {(B . n)}) by A5, A6, Lm6;

        then

         A7: ( rng A0) = ( rng B0) by A5, Lm7;

        

         A8: ( len A0) = (( dom A) /\ n) by RELAT_1: 61

        .= (( succ n) /\ n) by A5, Lm5

        .= n by Th1;

        

        thus A = (A0 ^ <%(A . n)%>) by A5, AFINSQ_1: 56

        .= (B0 ^ <%(A . n)%>) by A4, A7, A8

        .= (B0 ^ <%(B . n)%>) by A5, Lm7

        .= B by A6, AFINSQ_1: 56;

      end;

      

       A9: for n be Nat holds P[n] from NAT_1:sch 2( A1, A3);

      let A,B be decreasing Ordinal-Sequence;

      assume

       A10: ( rng A) = ( rng B);

      ( len A) is Nat;

      hence thesis by A9, A10;

    end;

    registration

      let a be Ordinal;

      cluster ( exp ( omega ,a)) -> Cantor-component;

      coherence

      proof

         0 in ( Segm 1) by CARD_1: 49, TARSKI:def 1;

        then (1 *^ ( exp ( omega ,a))) is Cantor-component by ORDINAL5:def 9;

        hence thesis by ORDINAL2: 39;

      end;

      let n be non zero Nat;

      cluster (n *^ ( exp ( omega ,a))) -> Cantor-component;

      coherence

      proof

         0 in ( Segm n) by ORDINAL3: 8;

        hence thesis by ORDINAL5:def 9;

      end;

    end

    registration

      cluster non zero -> Cantor-component for Nat;

      coherence

      proof

        let n be Nat;

        assume

         A1: n is non zero;

        n = (n *^ 1) by ORDINAL2: 39

        .= (n *^ ( exp ( omega qua Ordinal, 0 ))) by ORDINAL2: 43;

        hence thesis by A1;

      end;

    end

    registration

      let c be Cantor-component Ordinal;

      cluster <%c%> -> Cantor-normal-form;

      coherence

      proof

         A1:

        now

          let a be Ordinal;

          assume a in ( dom <%c%>);

          then a in ( Segm 1) by AFINSQ_1: 33;

          then a = 0 by CARD_1: 49, TARSKI:def 1;

          hence ( <%c%> . a) is Cantor-component;

        end;

        now

          let a,b be Ordinal;

          assume

           A2: a in b & b in ( dom <%c%>);

          then b in ( Segm 1) by AFINSQ_1: 33;

          hence ( omega -exponent ( <%c%> . b)) in ( omega -exponent ( <%c%> . a)) by A2, CARD_1: 49, TARSKI:def 1;

        end;

        hence thesis by A1, ORDINAL5:def 11;

      end;

    end

    theorem :: ORDINAL7:22

    

     Th35: for c,d be Cantor-component Ordinal st ( omega -exponent d) in ( omega -exponent c) holds <%c, d%> is Cantor-normal-form

    proof

      let c,d be Cantor-component Ordinal;

      assume ( omega -exponent d) in ( omega -exponent c);

      then ( omega -exponent ( <%d%> . 0 )) in ( omega -exponent ( last ( {} ^ <%c%>))) by AFINSQ_1: 92;

      then ( <%c%> ^ <%d%>) is Cantor-normal-form by Th33;

      hence thesis by AFINSQ_1:def 5;

    end;

    

     Lm8: for a be non empty Ordinal, n,m be non zero Nat holds <%(n *^ ( exp ( omega ,a))), m%> is Cantor-normal-form

    proof

      let a be non empty Ordinal, n,m be non zero Nat;

       0 c< n by XBOOLE_1: 2, XBOOLE_0:def 8;

      then 0 in n & n in omega by ORDINAL1: 11, ORDINAL1:def 12;

      then

       A1: ( omega -exponent (n *^ ( exp ( omega ,a)))) = a by ORDINAL5: 58;

      

       A2: ( omega -exponent m) = 0 by Th21, ORDINAL1:def 12;

       0 c< a by XBOOLE_1: 2, XBOOLE_0:def 8;

      hence thesis by A1, A2, Th35, ORDINAL1: 11;

    end;

    registration

      let a be non empty Ordinal, m be non zero Nat;

      cluster <%( exp ( omega ,a)), m%> -> Cantor-normal-form;

      coherence

      proof

        (1 *^ ( exp ( omega ,a))) = ( exp ( omega ,a)) by ORDINAL2: 39;

        hence thesis by Lm8;

      end;

      let n be non zero Nat;

      cluster <%(n *^ ( exp ( omega ,a))), m%> -> Cantor-normal-form;

      coherence by Lm8;

    end

    theorem :: ORDINAL7:23

    for c,d,e be Cantor-component Ordinal st ( omega -exponent d) in ( omega -exponent c) & ( omega -exponent e) in ( omega -exponent d) holds <%c, d, e%> is Cantor-normal-form

    proof

      let c,d,e be Cantor-component Ordinal;

      assume that

       A1: ( omega -exponent d) in ( omega -exponent c) and

       A2: ( omega -exponent e) in ( omega -exponent d);

      

       A3: <%d, e%> is Cantor-normal-form by A2, Th35;

      ( omega -exponent ( <%d, e%> . 0 )) in ( omega -exponent ( last ( {} ^ <%c%>))) by A1, AFINSQ_1: 92;

      then ( <%c%> ^ <%d, e%>) is Cantor-normal-form by A3, Th33;

      hence thesis by AFINSQ_1: 37;

    end;

    theorem :: ORDINAL7:24

    

     Th37: for A be non empty Cantor-normal-form Ordinal-Sequence holds for b be Ordinal, n be non zero Nat st b in ( omega -exponent ( last A)) holds (A ^ <%(n *^ ( exp ( omega ,b)))%>) is Cantor-normal-form

    proof

      let A be non empty Cantor-normal-form Ordinal-Sequence;

      let b be Ordinal, n be non zero Nat;

      assume

       A1: b in ( omega -exponent ( last A));

       0 c< n by XBOOLE_1: 2, XBOOLE_0:def 8;

      then 0 in n & n in omega by ORDINAL1: 11, ORDINAL1:def 12;

      then ( omega -exponent ( <%(n *^ ( exp ( omega ,b)))%> . 0 )) in ( omega -exponent ( last A)) by A1, ORDINAL5: 58;

      hence thesis by Th33;

    end;

    theorem :: ORDINAL7:25

    for A be non empty Cantor-normal-form Ordinal-Sequence holds for b be Ordinal, n be non zero Nat st ( omega -exponent ( last A)) <> 0 holds (A ^ <%n%>) is Cantor-normal-form

    proof

      let A be non empty Cantor-normal-form Ordinal-Sequence;

      let b be Ordinal, n be non zero Nat;

      assume ( omega -exponent ( last A)) <> 0 ;

      then 0 c< ( omega -exponent ( last A)) by XBOOLE_1: 2, XBOOLE_0:def 8;

      then

       A1: 0 in ( omega -exponent ( last A)) by ORDINAL1: 11;

      (A ^ <%(n *^ ( exp ( omega , 0 qua Ordinal)))%>) = (A ^ <%(n *^ 1)%>) by ORDINAL2: 43

      .= (A ^ <%n%>) by ORDINAL2: 39;

      hence thesis by A1, Th37;

    end;

    theorem :: ORDINAL7:26

    

     Th39: for A be non empty Cantor-normal-form Ordinal-Sequence holds for b be Ordinal, n be non zero Nat st ( omega -exponent (A . 0 )) in b holds ( <%(n *^ ( exp ( omega ,b)))%> ^ A) is Cantor-normal-form

    proof

      let A be non empty Cantor-normal-form Ordinal-Sequence;

      let b be Ordinal, n be non zero Nat;

      assume

       A1: ( omega -exponent (A . 0 )) in b;

       0 c< n by XBOOLE_1: 2, XBOOLE_0:def 8;

      then 0 in n & n in omega by ORDINAL1: 11, ORDINAL1:def 12;

      then ( omega -exponent (A . 0 )) in ( omega -exponent (n *^ ( exp ( omega ,b)))) by A1, ORDINAL5: 58;

      then ( omega -exponent (A . 0 )) in ( omega -exponent ( last ( {} ^ <%(n *^ ( exp ( omega ,b)))%>))) by AFINSQ_1: 92;

      hence thesis by Th33;

    end;

    theorem :: ORDINAL7:27

    

     Th40: for a1,a2,b be Ordinal st a1 in ( exp ( omega ,b)) & a2 in ( exp ( omega ,b)) holds (a1 +^ a2) in ( exp ( omega ,b))

    proof

      let a1,a2,b be Ordinal;

      assume

       A1: a1 in ( exp ( omega ,b)) & a2 in ( exp ( omega ,b));

      per cases ;

        suppose

         A2: 0 in a1 & 0 in a2;

        set d1 = ( omega -exponent a1), d2 = ( omega -exponent a2);

        consider n1 be Nat, c1 be Ordinal such that

         A3: a1 = ((n1 *^ ( exp ( omega ,d1))) +^ c1) & 0 in ( Segm n1) & c1 in ( exp ( omega ,d1)) by A2, ORDINAL5: 62;

        consider n2 be Nat, c2 be Ordinal such that

         A4: a2 = ((n2 *^ ( exp ( omega ,d2))) +^ c2) & 0 in ( Segm n2) & c2 in ( exp ( omega ,d2)) by A2, ORDINAL5: 62;

        

         A5: d1 in b

        proof

          assume not d1 in b;

          then

           A6: ( exp ( omega ,b)) c= ( exp ( omega ,d1)) by ORDINAL1: 16, ORDINAL4: 27;

          1 c= n1 by ORDINAL1: 21, Lm1, A3;

          then (1 *^ ( exp ( omega ,b))) c= (n1 *^ ( exp ( omega ,d1))) by A6, ORDINAL3: 20;

          then

           A7: ( exp ( omega ,b)) c= (n1 *^ ( exp ( omega ,d1))) by ORDINAL2: 39;

           0 c= c1;

          then (( exp ( omega ,b)) +^ 0 ) c= a1 by A3, A7, ORDINAL3: 18;

          then ( exp ( omega ,b)) c= a1 by ORDINAL2: 27;

          hence contradiction by A1, ORDINAL1: 5;

        end;

        

         A8: d2 in b

        proof

          assume not d2 in b;

          then

           A9: ( exp ( omega ,b)) c= ( exp ( omega ,d2)) by ORDINAL1: 16, ORDINAL4: 27;

          1 c= n2 by ORDINAL1: 21, Lm1, A4;

          then (1 *^ ( exp ( omega ,b))) c= (n2 *^ ( exp ( omega ,d2))) by A9, ORDINAL3: 20;

          then

           A10: ( exp ( omega ,b)) c= (n2 *^ ( exp ( omega ,d2))) by ORDINAL2: 39;

           0 c= c2;

          then (( exp ( omega ,b)) +^ 0 ) c= a2 by A4, A10, ORDINAL3: 18;

          then ( exp ( omega ,b)) c= a2 by ORDINAL2: 27;

          hence contradiction by A1, ORDINAL1: 5;

        end;

        a1 in ((n1 *^ ( exp ( omega ,d1))) +^ ( exp ( omega ,d1))) by A3, ORDINAL2: 32;

        then

         A11: a1 in (( succ n1) *^ ( exp ( omega ,d1))) by ORDINAL2: 36;

        a2 in ((n2 *^ ( exp ( omega ,d2))) +^ ( exp ( omega ,d2))) by A4, ORDINAL2: 32;

        then

         A12: a2 in (( succ n2) *^ ( exp ( omega ,d2))) by ORDINAL2: 36;

        per cases by ORDINAL1: 16;

          suppose d1 c= d2;

          then ( exp ( omega ,d1)) c= ( exp ( omega ,d2)) by ORDINAL4: 27;

          then (( succ n1) *^ ( exp ( omega ,d1))) c= (( succ n1) *^ ( exp ( omega ,d2))) by ORDINAL2: 42;

          then (a1 +^ a2) in ((( succ n1) *^ ( exp ( omega ,d2))) +^ (( succ n2) *^ ( exp ( omega ,d2)))) by A11, A12, ORDINAL3: 17;

          then

           A13: (a1 +^ a2) in ((( succ n1) +^ ( succ n2)) *^ ( exp ( omega ,d2))) by ORDINAL3: 46;

          ((( succ n1) +^ ( succ n2)) *^ ( exp ( omega ,d2))) in ( exp ( omega ,b)) by A8, ORDINAL5: 7;

          hence thesis by A13, ORDINAL1: 10;

        end;

          suppose d2 in d1;

          then ( exp ( omega ,d2)) c= ( exp ( omega ,d1)) by ORDINAL1:def 2, ORDINAL4: 27;

          then (( succ n2) *^ ( exp ( omega ,d2))) c= (( succ n2) *^ ( exp ( omega ,d1))) by ORDINAL2: 42;

          then (a1 +^ a2) in ((( succ n1) *^ ( exp ( omega ,d1))) +^ (( succ n2) *^ ( exp ( omega ,d1)))) by A11, A12, ORDINAL3: 17;

          then

           A14: (a1 +^ a2) in ((( succ n1) +^ ( succ n2)) *^ ( exp ( omega ,d1))) by ORDINAL3: 46;

          ((( succ n1) +^ ( succ n2)) *^ ( exp ( omega ,d1))) in ( exp ( omega ,b)) by A5, ORDINAL5: 7;

          hence thesis by A14, ORDINAL1: 10;

        end;

      end;

        suppose not 0 in a1;

        then a1 = 0 by ORDINAL1: 16, XBOOLE_1: 3;

        hence thesis by A1, ORDINAL2: 30;

      end;

        suppose not 0 in a2;

        then a2 = 0 by ORDINAL1: 16, XBOOLE_1: 3;

        hence thesis by A1, ORDINAL2: 27;

      end;

    end;

    theorem :: ORDINAL7:28

    

     Th41: for A be finite Ordinal-Sequence, b be Ordinal st for a be Ordinal st a in ( dom A) holds (A . a) in ( exp ( omega ,b)) holds ( Sum^ A) in ( exp ( omega ,b))

    proof

      defpred P[ Nat] means for A be finite Ordinal-Sequence, b be Ordinal st ( dom A) = $1 & for a be Ordinal st a in ( dom A) holds (A . a) in ( exp ( omega ,b)) holds ( Sum^ A) in ( exp ( omega ,b));

      

       A1: P[ 0 ]

      proof

        let A be finite Ordinal-Sequence, b be Ordinal;

        assume that

         A2: ( dom A) = 0 and for a be Ordinal st a in ( dom A) holds (A . a) in ( exp ( omega ,b));

        A = {} by A2;

        then ( Sum^ A) in 1 by ORDINAL5: 52, CARD_1: 49, TARSKI:def 1;

        then

         A3: ( Sum^ A) in ( exp ( omega , 0 qua Ordinal)) by ORDINAL2: 43;

        per cases ;

          suppose 0 in b;

          then ( exp ( omega , 0 qua Ordinal)) in ( exp ( omega ,b)) by ORDINAL4: 24;

          hence thesis by A3, ORDINAL1: 10;

        end;

          suppose not 0 in b;

          hence thesis by A3, ORDINAL1: 16, XBOOLE_1: 3;

        end;

      end;

      

       A4: for n be Nat st P[n] holds P[(n + 1)]

      proof

        let n be Nat;

        assume

         A5: P[n];

        let A be finite Ordinal-Sequence, b be Ordinal;

        assume that

         A6: ( dom A) = (n + 1) and

         A7: for a be Ordinal st a in ( dom A) holds (A . a) in ( exp ( omega ,b));

        A <> {} by A6;

        then

        consider A0 be XFinSequence, a0 be object such that

         A8: A = (A0 ^ <%a0%>) by AFINSQ_1: 40;

        consider c be Ordinal such that

         A9: ( rng A) c= c by ORDINAL2:def 4;

        ( rng A0) c= ( rng A) by A8, AFINSQ_1: 24;

        then

        reconsider A0 as finite Ordinal-Sequence by A9, XBOOLE_1: 1, ORDINAL2:def 4;

        ( rng <%a0%>) c= ( rng A) by A8, AFINSQ_1: 25;

        then {a0} c= ( rng A) by AFINSQ_1: 33;

        then a0 in ( rng A) by ZFMISC_1: 31;

        then

        reconsider a0 as Ordinal;

        

         A10: (( len A0) + 1) = (n + 1) by A6, A8, AFINSQ_1: 75;

        now

          let a be Ordinal;

          assume

           A11: a in ( dom A0);

          then

           A12: (A0 . a) = (A . a) by A8, AFINSQ_1:def 3;

          ( dom A0) c= ( dom A) by A8, AFINSQ_1: 21;

          hence (A0 . a) in ( exp ( omega ,b)) by A7, A11, A12;

        end;

        then

         A13: ( Sum^ A0) in ( exp ( omega ,b)) by A5, A10;

        (n + 0 ) < (n + 1) by XREAL_1: 8;

        then (A . n) in ( exp ( omega ,b)) by A7, AFINSQ_1: 86, A6;

        then

         A14: a0 in ( exp ( omega ,b)) by A8, A10, AFINSQ_1: 36;

        ( Sum^ A) = (( Sum^ A0) +^ a0) by A8, ORDINAL5: 54;

        hence thesis by A13, A14, Th40;

      end;

      

       A15: for n be Nat holds P[n] from NAT_1:sch 2( A1, A4);

      let A be finite Ordinal-Sequence, b be Ordinal;

      thus thesis by A15;

    end;

    theorem :: ORDINAL7:29

    

     Th42: for a,b be Ordinal, n be Nat st a in ( exp ( omega ,b)) holds (n *^ a) in ( exp ( omega ,b))

    proof

      let a,b be Ordinal, n be Nat;

      assume a in ( exp ( omega ,b));

      then for c be Ordinal st c in ( dom (n --> a)) holds ((n --> a) . c) in ( exp ( omega ,b)) by FUNCOP_1: 7;

      then ( Sum^ (n --> a)) in ( exp ( omega ,b)) by Th41;

      hence thesis by Th26;

    end;

    theorem :: ORDINAL7:30

    

     Th43: for A be finite Ordinal-Sequence, a be Ordinal st ( <%a%> ^ A) is Cantor-normal-form holds ( Sum^ A) in ( exp ( omega ,( omega -exponent a)))

    proof

      let A be finite Ordinal-Sequence, a be Ordinal;

      assume ( <%a%> ^ A) is Cantor-normal-form;

      then

      reconsider B = ( <%a%> ^ A) as Cantor-normal-form Ordinal-Sequence;

      now

        let c be Ordinal;

        assume

         A1: c in ( dom A);

        then

        reconsider n = c as Nat;

        (( len <%a%>) + n) in ( dom B) by A1, AFINSQ_1: 23;

        then

         A2: (n + 1) in ( dom B) by AFINSQ_1: 34;

        (B . (( len <%a%>) + n)) = (A . n) by A1, AFINSQ_1:def 3;

        then

         A3: (A . n) = (B . (n + 1)) by AFINSQ_1: 34;

         0 in ( Segm (n + 1)) by NAT_1: 44;

        then ( omega -exponent (B . (n + 1))) in ( omega -exponent (B . 0 )) by A2, ORDINAL5:def 11;

        then ( exp ( omega ,( omega -exponent (A . n)))) in ( exp ( omega ,( omega -exponent (B . 0 )))) by A3, ORDINAL4: 24;

        then

         A4: ( exp ( omega ,( omega -exponent (A . n)))) in ( exp ( omega ,( omega -exponent a))) by AFINSQ_1: 35;

        (B . (n + 1)) is Cantor-component by A2, ORDINAL5:def 11;

        then

        consider b be Ordinal, m be Nat such that

         A5: 0 in ( Segm m) & (A . n) = (m *^ ( exp ( omega ,b))) by A3, ORDINAL5:def 9;

         0 in m & m in omega by A5, ORDINAL1:def 12;

        then ( omega -exponent (A . n)) = b by A5, ORDINAL5: 58;

        hence (A . c) in ( exp ( omega ,( omega -exponent a))) by A4, A5, Th42;

      end;

      hence ( Sum^ A) in ( exp ( omega ,( omega -exponent a))) by Th41;

    end;

    theorem :: ORDINAL7:31

    

     Th44: for A be Cantor-normal-form Ordinal-Sequence holds ( omega -exponent ( Sum^ A)) = ( omega -exponent (A . 0 ))

    proof

      defpred P[ Nat] means for A be Cantor-normal-form Ordinal-Sequence st ( len A) = $1 holds ( omega -exponent ( Sum^ A)) = ( omega -exponent (A . 0 ));

      

       A1: P[ 0 ]

      proof

        let A be Cantor-normal-form Ordinal-Sequence;

        assume ( len A) = 0 ;

        then A = {} ;

        hence ( omega -exponent ( Sum^ A)) = ( omega -exponent (A . 0 )) by ORDINAL5: 52;

      end;

      

       A3: for n be Nat st P[n] holds P[(n + 1)]

      proof

        let n be Nat;

        assume

         A4: P[n];

        let A be Cantor-normal-form Ordinal-Sequence;

        assume

         A5: ( len A) = (n + 1);

        then A <> {} ;

        then

        consider c be Cantor-component Ordinal, B be Cantor-normal-form Ordinal-Sequence such that

         A6: A = ( <%c%> ^ B) by ORDINAL5: 67;

        per cases ;

          suppose

           A7: B = {} ;

          ( Sum^ A) = (c +^ ( Sum^ B)) by A6, ORDINAL5: 55

          .= (A . 0 ) by A6, A7, ORDINAL5: 52, ORDINAL2: 27;

          hence thesis;

        end;

          suppose

           A8: B <> {} ;

          then {} c< ( dom B) by XBOOLE_1: 2, XBOOLE_0:def 8;

          then

           A9: 0 in ( dom B) by ORDINAL1: 11;

          (n + 1) = (( len <%c%>) + ( len B)) by A5, A6, AFINSQ_1: 17

          .= (( len B) + 1) by AFINSQ_1: 34;

          then

           A10: ( omega -exponent ( Sum^ B)) = ( omega -exponent (B . 0 )) by A4;

          (A . (( len <%c%>) + 0 )) = (B . 0 ) by A6, A9, AFINSQ_1:def 3;

          then

           A11: (A . 1) = (B . 0 ) by AFINSQ_1: 34;

          (( len <%c%>) + 0 ) in ( dom A) by A6, A9, AFINSQ_1: 23;

          then

           A12: 1 in ( dom A) by AFINSQ_1: 34;

           0 in 1 by CARD_1: 49, TARSKI:def 1;

          then

           A13: ( omega -exponent ( Sum^ B)) in ( omega -exponent (A . 0 )) by A10, A11, A12, ORDINAL5:def 11;

          

           A14: ( omega -exponent (A . 0 )) c= ( omega -exponent ( Sum^ A)) by Th22, ORDINAL5: 56;

          consider d be Ordinal, m be Nat such that

           A15: 0 in ( Segm m) & c = (m *^ ( exp ( omega ,d))) by ORDINAL5:def 9;

           0 in m & m in omega by A15, ORDINAL1:def 12;

          then ( omega -exponent c) = d by A15, ORDINAL5: 58;

          then

           A16: ( omega -exponent (A . 0 )) = d by A6, AFINSQ_1: 35;

          assume ( omega -exponent ( Sum^ A)) <> ( omega -exponent (A . 0 ));

          then ( omega -exponent (A . 0 )) in ( omega -exponent ( Sum^ A)) by A14, XBOOLE_0:def 8, ORDINAL1: 11;

          then

           A17: ( exp ( omega ,d)) in ( exp ( omega ,( omega -exponent ( Sum^ A)))) by A16, ORDINAL4: 24;

          then

           A18: c in ( exp ( omega ,( omega -exponent ( Sum^ A)))) by A15, Th42;

          set e = ( omega -exponent ( Sum^ B));

          

           A19: 0 in ( Sum^ B)

          proof

            assume not 0 in ( Sum^ B);

            then ( Sum^ B) c= 0 by ORDINAL1: 16;

            hence contradiction by A8;

          end;

          

           A20: ( Sum^ B) in ( exp ( omega ,( succ e)))

          proof

            assume not ( Sum^ B) in ( exp ( omega ,( succ e)));

            then ( exp ( omega ,( succ e))) c= ( Sum^ B) by ORDINAL1: 16;

            then ( succ e) c= e by A19, ORDINAL5:def 10;

            hence contradiction by ORDINAL1: 5, ORDINAL1: 6;

          end;

          ( exp ( omega ,( succ e))) c= ( exp ( omega ,d)) by A13, A16, ORDINAL1: 21, ORDINAL4: 27;

          then ( Sum^ B) in ( exp ( omega ,( omega -exponent ( Sum^ A)))) by A17, A20, ORDINAL1: 10;

          then (c +^ ( Sum^ B)) in ( exp ( omega ,( omega -exponent ( Sum^ A)))) by A18, Th40;

          then

           A22: ( Sum^ A) in ( exp ( omega ,( omega -exponent ( Sum^ A)))) by A6, ORDINAL5: 55;

          ( Sum^ B) c= (c +^ ( Sum^ B)) by ORDINAL3: 24;

          then ( Sum^ B) c= ( Sum^ A) by A6, ORDINAL5: 55;

          then ( exp ( omega ,( omega -exponent ( Sum^ A)))) c= ( Sum^ A) by A19, ORDINAL5:def 10;

          then ( Sum^ A) in ( Sum^ A) by A22;

          hence contradiction;

        end;

      end;

      

       A23: for n be Nat holds P[n] from NAT_1:sch 2( A1, A3);

      let A be Cantor-normal-form Ordinal-Sequence;

      ( len A) is Nat;

      hence thesis by A23;

    end;

    theorem :: ORDINAL7:32

    

     Th45: for A,B be Cantor-normal-form Ordinal-Sequence st ( Sum^ A) = ( Sum^ B) holds A = B

    proof

      defpred P[ Nat] means for A,B be Cantor-normal-form Ordinal-Sequence st (( dom A) \/ ( dom B)) = $1 & ( Sum^ A) = ( Sum^ B) holds A = B;

      

       A1: P[ 0 ]

      proof

        let A,B be Cantor-normal-form Ordinal-Sequence;

        assume (( dom A) \/ ( dom B)) = 0 & ( Sum^ A) = ( Sum^ B);

        then A is empty & B is empty;

        hence thesis;

      end;

      

       A2: for n be Nat st P[n] holds P[(n + 1)]

      proof

        let n be Nat;

        assume

         A3: P[n];

        let A,B be Cantor-normal-form Ordinal-Sequence;

        assume

         A4: (( dom A) \/ ( dom B)) = (n + 1) & ( Sum^ A) = ( Sum^ B);

        ( dom A) <> {}

        proof

          assume

           A5: ( dom A) = {} ;

          then A is empty;

          then B = {} by A4, ORDINAL5: 52;

          hence contradiction by A4, A5;

        end;

        then

         A6: A <> {} ;

        ( dom B) <> {}

        proof

          assume ( dom B) = {} ;

          then B is empty;

          hence contradiction by A4, A6, ORDINAL5: 52;

        end;

        then B <> {} ;

        then

        consider b be Cantor-component Ordinal, B0 be Cantor-normal-form Ordinal-Sequence such that

         A7: B = ( <%b%> ^ B0) by ORDINAL5: 67;

        consider a be Cantor-component Ordinal, A0 be Cantor-normal-form Ordinal-Sequence such that

         A8: A = ( <%a%> ^ A0) by A6, ORDINAL5: 67;

        

         A9: (a +^ ( Sum^ A0)) = ( Sum^ B) by A4, A8, ORDINAL5: 55

        .= (b +^ ( Sum^ B0)) by A7, ORDINAL5: 55;

        

         A10: a = b

        proof

          

           A11: (A . 0 ) = a & (B . 0 ) = b by A7, A8, AFINSQ_1: 35;

          

          then

           A12: ( omega -exponent a) = ( omega -exponent ( Sum^ B)) by A4, Th44

          .= ( omega -exponent b) by A11, Th44;

          consider d1 be Ordinal, n1 be Nat such that

           A13: 0 in ( Segm n1) & a = (n1 *^ ( exp ( omega ,d1))) by ORDINAL5:def 9;

          consider d2 be Ordinal, n2 be Nat such that

           A14: 0 in ( Segm n2) & b = (n2 *^ ( exp ( omega ,d2))) by ORDINAL5:def 9;

           0 in n1 & n1 in omega by A13, ORDINAL1:def 12;

          then

           A15: ( omega -exponent a) = d1 by A13, ORDINAL5: 58;

           0 in n2 & n2 in omega by A14, ORDINAL1:def 12;

          then

           A16: ( omega -exponent b) = d2 by A14, ORDINAL5: 58;

          then

           A17: d1 = d2 by A12, A15;

          assume a <> b;

          per cases by ORDINAL1: 14;

            suppose

             A18: a in b;

            

            then (a +^ ( Sum^ A0)) = ((a +^ (b -^ a)) +^ ( Sum^ B0)) by A9, ORDINAL3: 51

            .= (a +^ ((b -^ a) +^ ( Sum^ B0))) by ORDINAL3: 30;

            then

             A19: ( Sum^ A0) = ((b -^ a) +^ ( Sum^ B0)) by ORDINAL3: 21;

            

             A20: n1 in n2 by A13, A14, A17, A18, ORDINAL3: 34;

            

             A21: (b -^ a) = ((n2 -^ n1) *^ ( exp ( omega ,d1))) by A13, A14, A17, ORDINAL3: 63;

             0 in (n2 -^ n1) & (n2 -^ n1) in omega by A20, ORDINAL3: 55, ORDINAL1:def 12;

            then

             A22: ( omega -exponent (b -^ a)) = d1 by A21, ORDINAL5: 58;

            

             A23: (b -^ a) c= ((b -^ a) +^ ( Sum^ B0)) by ORDINAL3: 24;

            then

             A24: d1 c= ( omega -exponent ( Sum^ A0)) by A19, A22, Th22;

             0 in (b -^ a) by A18, ORDINAL3: 55;

            then

             A25: 0 in ( Sum^ A0) by A19, A23;

            ( Sum^ A0) in ( exp ( omega ,( omega -exponent a))) by A8, Th43;

            hence contradiction by A15, A24, A25, Th23, ORDINAL1: 5;

          end;

            suppose

             A26: b in a;

            

            then (b +^ ( Sum^ B0)) = ((b +^ (a -^ b)) +^ ( Sum^ A0)) by A9, ORDINAL3: 51

            .= (b +^ ((a -^ b) +^ ( Sum^ A0))) by ORDINAL3: 30;

            then

             A27: ( Sum^ B0) = ((a -^ b) +^ ( Sum^ A0)) by ORDINAL3: 21;

            

             A28: n2 in n1 by A13, A14, A17, A26, ORDINAL3: 34;

            

             A29: (a -^ b) = ((n1 -^ n2) *^ ( exp ( omega ,d1))) by A13, A14, A17, ORDINAL3: 63;

             0 in (n1 -^ n2) & (n1 -^ n2) in omega by A28, ORDINAL3: 55, ORDINAL1:def 12;

            then

             A30: ( omega -exponent (a -^ b)) = d1 by A29, ORDINAL5: 58;

            

             A31: (a -^ b) c= ((a -^ b) +^ ( Sum^ A0)) by ORDINAL3: 24;

            then

             A32: d1 c= ( omega -exponent ( Sum^ B0)) by A27, A30, Th22;

             0 in (a -^ b) by A26, ORDINAL3: 55;

            then

             A33: 0 in ( Sum^ B0) by A27, A31;

            ( Sum^ B0) in ( exp ( omega ,( omega -exponent b))) by A7, Th43;

            hence contradiction by A16, A17, A32, A33, Th23, ORDINAL1: 5;

          end;

        end;

        then

         A34: ( Sum^ A0) = ( Sum^ B0) by A9, ORDINAL3: 21;

        (( dom A0) \/ ( dom B0)) = ((( max (( len A0),( len B0))) + 1) - 1)

        .= (( max ((( len A0) + 1),(( len B0) + 1))) - 1) by FUZZY_2: 42

        .= (( max ((( len A0) + ( len <%a%>)),(( len B0) + 1))) - 1) by AFINSQ_1: 34

        .= (( max ((( len A0) + ( len <%a%>)),(( len B0) + ( len <%b%>)))) - 1) by AFINSQ_1: 34

        .= (( max (( len A),(( len B0) + ( len <%b%>)))) - 1) by A8, AFINSQ_1: 17

        .= (( max (( len A),( len B))) - 1) by A7, AFINSQ_1: 17

        .= n by A4;

        hence thesis by A3, A7, A8, A10, A34;

      end;

      

       A35: for n be Nat holds P[n] from NAT_1:sch 2( A1, A2);

      let A,B be Cantor-normal-form Ordinal-Sequence;

      assume

       A36: ( Sum^ A) = ( Sum^ B);

      (( dom A) \/ ( dom B)) is natural;

      hence thesis by A35, A36;

    end;

    definition

      let A be Ordinal-Sequence, b be Ordinal;

      :: ORDINAL7:def1

      func b -exponent A -> Ordinal-Sequence means

      : Def1: ( dom it ) = ( dom A) & for a be object st a in ( dom A) holds (it . a) = (b -exponent (A . a));

      existence

      proof

        deffunc F( object) = (b -exponent (A . $1));

        consider f be Function such that

         A1: ( dom f) = ( dom A) & for a be object st a in ( dom A) holds (f . a) = F(a) from FUNCT_1:sch 3;

        reconsider f as Sequence by A1, ORDINAL1: 31;

        now

          reconsider c = ( sup ( rng f)) as Ordinal;

          take c;

          now

            let y be object;

            assume

             A2: y in ( rng f);

            then

            consider x be object such that

             A3: x in ( dom f) & (f . x) = y by FUNCT_1:def 3;

            (f . x) = (b -exponent (A . x)) by A1, A3;

            hence y in ( sup ( rng f)) by A2, A3, ORDINAL2: 19;

          end;

          hence ( rng f) c= ( sup ( rng f)) by TARSKI:def 3;

        end;

        then

        reconsider f as Ordinal-Sequence by ORDINAL2:def 4;

        take f;

        thus thesis by A1;

      end;

      uniqueness

      proof

        let f1,f2 be Ordinal-Sequence;

        assume that

         A4: ( dom f1) = ( dom A) and

         A5: for a be object st a in ( dom A) holds (f1 . a) = (b -exponent (A . a)) and

         A6: ( dom f2) = ( dom A) and

         A7: for a be object st a in ( dom A) holds (f2 . a) = (b -exponent (A . a));

        now

          let a be object;

          assume

           A8: a in ( dom f1);

          

          hence (f1 . a) = (b -exponent (A . a)) by A4, A5

          .= (f2 . a) by A4, A7, A8;

        end;

        hence thesis by A4, A6, FUNCT_1: 2;

      end;

    end

    registration

      let A be empty Ordinal-Sequence, b be Ordinal;

      cluster (b -exponent A) -> empty;

      coherence

      proof

        ( dom A) = ( dom (b -exponent A)) by Def1;

        hence thesis;

      end;

    end

    registration

      let A be non empty Ordinal-Sequence, b be Ordinal;

      cluster (b -exponent A) -> non empty;

      coherence

      proof

        ( dom A) = ( dom (b -exponent A)) by Def1;

        hence thesis;

      end;

    end

    registration

      let A be finite Ordinal-Sequence, b be Ordinal;

      cluster (b -exponent A) -> finite;

      coherence

      proof

        ( dom A) = ( dom (b -exponent A)) by Def1;

        hence thesis by FINSET_1: 10;

      end;

    end

    registration

      let A be infinite Ordinal-Sequence, b be Ordinal;

      cluster (b -exponent A) -> infinite;

      coherence

      proof

        ( dom A) = ( dom (b -exponent A)) by Def1;

        hence thesis by FINSET_1: 10;

      end;

    end

    theorem :: ORDINAL7:33

    

     Th46: for a,b be Ordinal holds (b -exponent <%a%>) = <%(b -exponent a)%>

    proof

      let a,b be Ordinal;

      

       A1: ( dom (b -exponent <%a%>)) = ( dom <%a%>) by Def1

      .= 1 by AFINSQ_1:def 4;

       0 in 1 by TARSKI:def 1, CARD_1: 49;

      then 0 in ( dom <%a%>) by AFINSQ_1:def 4;

      

      then ((b -exponent <%a%>) . 0 ) = (b -exponent ( <%a%> . 0 )) by Def1

      .= (b -exponent a);

      hence thesis by A1, AFINSQ_1:def 4;

    end;

    theorem :: ORDINAL7:34

    

     Th47: for A,B be Ordinal-Sequence, b be Ordinal holds (b -exponent (A ^ B)) = ((b -exponent A) ^ (b -exponent B))

    proof

      let A,B be Ordinal-Sequence, b be Ordinal;

      

       A1: ( dom (b -exponent (A ^ B))) = ( dom (A ^ B)) by Def1

      .= (( dom A) +^ ( dom B)) by ORDINAL4:def 1

      .= (( dom A) +^ ( dom (b -exponent B))) by Def1

      .= (( dom (b -exponent A)) +^ ( dom (b -exponent B))) by Def1

      .= ( dom ((b -exponent A) ^ (b -exponent B))) by ORDINAL4:def 1;

      now

        let x be object;

        assume x in ( dom (b -exponent (A ^ B)));

        then

         A2: x in ( dom (A ^ B)) by Def1;

        then

         A3: ((b -exponent (A ^ B)) . x) = (b -exponent ((A ^ B) . x)) by Def1;

        reconsider c = x as Ordinal by A2;

        c in ( dom A) or (( dom A) c= c & (c -^ ( dom A)) in ( dom B))

        proof

          assume not c in ( dom A);

          hence

           A4: ( dom A) c= c by ORDINAL1: 16;

          c in (( dom A) +^ ( dom B)) by A2, ORDINAL4:def 1;

          then (c -^ ( dom A)) in ((( dom A) +^ ( dom B)) -^ ( dom A)) by A4, ORDINAL3: 53;

          hence thesis by ORDINAL3: 52;

        end;

        per cases ;

          suppose

           A5: c in ( dom A);

          then

           A6: c in ( dom (b -exponent A)) by Def1;

          ((A ^ B) . x) = (A . x) by A5, ORDINAL4:def 1;

          

          hence ((b -exponent (A ^ B)) . x) = ((b -exponent A) . x) by A3, A5, Def1

          .= (((b -exponent A) ^ (b -exponent B)) . x) by A6, ORDINAL4:def 1;

        end;

          suppose

           A7: ( dom A) c= c & (c -^ ( dom A)) in ( dom B);

          then

           A8: (c -^ ( dom A)) in ( dom (b -exponent B)) by Def1;

          ((A ^ B) . x) = ((A ^ B) . (( dom A) +^ (c -^ ( dom A)))) by A7, ORDINAL3:def 5

          .= (B . (c -^ ( dom A))) by A7, ORDINAL4:def 1;

          

          hence ((b -exponent (A ^ B)) . x) = ((b -exponent B) . (c -^ ( dom A))) by A3, A7, Def1

          .= (((b -exponent A) ^ (b -exponent B)) . (( dom (b -exponent A)) +^ (c -^ ( dom A)))) by A8, ORDINAL4:def 1

          .= (((b -exponent A) ^ (b -exponent B)) . (( dom A) +^ (c -^ ( dom A)))) by Def1

          .= (((b -exponent A) ^ (b -exponent B)) . x) by A7, ORDINAL3:def 5;

        end;

      end;

      hence thesis by A1, FUNCT_1: 2;

    end;

    theorem :: ORDINAL7:35

    

     Th48: for A be Ordinal-Sequence, b,c be Ordinal holds (b -exponent (A | c)) = ((b -exponent A) | c)

    proof

      let A be Ordinal-Sequence, b,c be Ordinal;

      

       A1: ( dom (b -exponent (A | c))) = ( dom (A | c)) by Def1

      .= (( dom A) /\ c) by RELAT_1: 61

      .= (( dom (b -exponent A)) /\ c) by Def1

      .= ( dom ((b -exponent A) | c)) by RELAT_1: 61;

      now

        let x be object;

        assume

         A2: x in ( dom (b -exponent (A | c)));

        then

         A3: x in ( dom (A | c)) by Def1;

        then

         A4: x in ( dom A) by RELAT_1: 57;

        

        thus ((b -exponent (A | c)) . x) = (b -exponent ((A | c) . x)) by A3, Def1

        .= (b -exponent (A . x)) by A3, FUNCT_1: 47

        .= ((b -exponent A) . x) by A4, Def1

        .= (((b -exponent A) | c) . x) by A1, A2, FUNCT_1: 47;

      end;

      hence thesis by A1, FUNCT_1: 2;

    end;

    theorem :: ORDINAL7:36

    

     Th49: for A be finite Ordinal-Sequence, b be Ordinal, n be Nat holds (b -exponent (A /^ n)) = ((b -exponent A) /^ n)

    proof

      let A be finite Ordinal-Sequence, b be Ordinal, n be Nat;

      

       A1: ( dom (b -exponent (A /^ n))) = ( len (A /^ n)) by Def1

      .= (( len A) -' n) by AFINSQ_2:def 2

      .= (( len (b -exponent A)) -' n) by Def1

      .= ( dom ((b -exponent A) /^ n)) by AFINSQ_2:def 2;

      now

        let k be Nat;

        assume

         A2: k in ( dom (b -exponent (A /^ n)));

        then

         A3: k in ( dom (A /^ n)) by Def1;

        

         A4: (b -exponent (A . (k + n))) = ((b -exponent A) . (k + n))

        proof

          per cases ;

            suppose (k + n) in ( dom A);

            hence thesis by Def1;

          end;

            suppose

             A5: not (k + n) in ( dom A);

            then (A . (k + n)) = {} by FUNCT_1:def 2;

            then

             A6: (b -exponent (A . (k + n))) = {} by ORDINAL5:def 10;

             not (k + n) in ( dom (b -exponent A)) by A5, Def1;

            hence thesis by A6, FUNCT_1:def 2;

          end;

        end;

        

        thus ((b -exponent (A /^ n)) . k) = (b -exponent ((A /^ n) . k)) by A3, Def1

        .= (b -exponent (A . (k + n))) by A3, AFINSQ_2:def 2

        .= (((b -exponent A) /^ n) . k) by A1, A2, A4, AFINSQ_2:def 2;

      end;

      hence thesis by A1, AFINSQ_1: 8;

    end;

    registration

      let A be Cantor-normal-form Ordinal-Sequence;

      cluster ( omega -exponent A) -> decreasing;

      coherence

      proof

        now

          let a,b be Ordinal;

          assume

           A1: a in b & b in ( dom ( omega -exponent A));

          then

           A2: b in ( dom A) by Def1;

          then (( omega -exponent A) . a) = ( omega -exponent (A . a)) & (( omega -exponent A) . b) = ( omega -exponent (A . b)) by A1, Def1, ORDINAL1: 10;

          hence (( omega -exponent A) . b) in (( omega -exponent A) . a) by A1, A2, ORDINAL5:def 11;

        end;

        hence thesis by ORDINAL5:def 1;

      end;

    end

    theorem :: ORDINAL7:37

    for A,B be Ordinal-Sequence st (A ^ B) is Cantor-normal-form holds ( rng ( omega -exponent A)) misses ( rng ( omega -exponent B))

    proof

      let A,B be Ordinal-Sequence;

      assume

       A1: (A ^ B) is Cantor-normal-form;

      (( rng ( omega -exponent A)) /\ ( rng ( omega -exponent B))) = {}

      proof

        assume (( rng ( omega -exponent A)) /\ ( rng ( omega -exponent B))) <> {} ;

        then

        consider y be object such that

         A2: y in (( rng ( omega -exponent A)) /\ ( rng ( omega -exponent B))) by XBOOLE_0:def 1;

        

         A3: y in ( rng ( omega -exponent A)) & y in ( rng ( omega -exponent B)) by A2, XBOOLE_0:def 4;

        then

        consider x1 be object such that

         A4: x1 in ( dom ( omega -exponent A)) & (( omega -exponent A) . x1) = y by FUNCT_1:def 3;

        consider x2 be object such that

         A5: x2 in ( dom ( omega -exponent B)) & (( omega -exponent B) . x2) = y by A3, FUNCT_1:def 3;

        reconsider x1, x2 as Ordinal by A4, A5;

        

         A6: x1 in ( dom A) by A4, Def1;

        then

         A7: (A . x1) = ((A ^ B) . x1) by ORDINAL4:def 1;

        

         A8: x2 in ( dom B) by A5, Def1;

        then

         A9: (B . x2) = ((A ^ B) . (( dom A) +^ x2)) by ORDINAL4:def 1;

        ( dom A) c= (( dom A) +^ x2) by ORDINAL3: 24;

        then

         A10: x1 in (( dom A) +^ x2) by A6;

        (( dom A) +^ x2) in (( dom A) +^ ( dom B)) by A8, ORDINAL2: 32;

        then (( dom A) +^ x2) in ( dom (A ^ B)) by ORDINAL4:def 1;

        then ( omega -exponent ((A ^ B) . (( dom A) +^ x2))) in ( omega -exponent ((A ^ B) . x1)) by A1, A10, ORDINAL5:def 11;

        then (( omega -exponent B) . x2) in ( omega -exponent (A . x1)) by A7, A8, A9, Def1;

        then (( omega -exponent B) . x2) in (( omega -exponent A) . x1) by A6, Def1;

        hence contradiction by A4, A5;

      end;

      hence thesis by XBOOLE_0:def 7;

    end;

    theorem :: ORDINAL7:38

    

     Th51: for A be Cantor-normal-form Ordinal-Sequence holds 0 in ( rng ( omega -exponent A)) iff A <> {} & ( omega -exponent ( last A)) = 0

    proof

      let A be Cantor-normal-form Ordinal-Sequence;

      hereby

        assume 0 in ( rng ( omega -exponent A));

        then

        consider x be object such that

         A1: x in ( dom ( omega -exponent A)) & (( omega -exponent A) . x) = 0 by FUNCT_1:def 3;

        thus

         A2: A <> {} by A1;

        

         A3: x in ( dom A) by A1, Def1;

        then ( omega -exponent ( last A)) c= ( omega -exponent (A . x)) by A2, Th31;

        then ( omega -exponent ( last A)) c= 0 by A1, A3, Def1;

        hence ( omega -exponent ( last A)) = 0 ;

      end;

      assume

       A4: A <> {} & ( omega -exponent ( last A)) = 0 ;

      then

      consider A0 be Cantor-normal-form Ordinal-Sequence, a0 be Cantor-component Ordinal such that

       A5: A = (A0 ^ <%a0%>) by Th29;

       0 in 1 by CARD_1: 49, TARSKI:def 1;

      then 0 in ( dom <%a0%>) by AFINSQ_1: 33;

      then

       A6: (( len A0) + 0 ) in ( dom A) by A5, AFINSQ_1: 23;

      then

       A7: ( len A0) in ( dom ( omega -exponent A)) by Def1;

       0 = ( omega -exponent a0) by A4, A5, AFINSQ_1: 92

      .= ( omega -exponent (A . ( len A0))) by A5, AFINSQ_1: 36

      .= (( omega -exponent A) . ( len A0)) by A6, Def1;

      hence thesis by A7, FUNCT_1: 3;

    end;

    definition

      let a,b be Ordinal;

      :: ORDINAL7:def2

      func b -leading_coeff a -> Ordinal equals (a div^ ( exp (b,(b -exponent a))));

      coherence ;

    end

    theorem :: ORDINAL7:39

    

     Th52: for a be Ordinal holds ( 0 -leading_coeff a) = a

    proof

      let a be Ordinal;

      

      thus ( 0 -leading_coeff a) = (a div^ ( exp ( 0 qua Ordinal, 0 ))) by ORDINAL5:def 10

      .= (a div^ 1) by ORDINAL2: 43

      .= a by ORDINAL3: 71;

    end;

    theorem :: ORDINAL7:40

    

     Th53: for a be Ordinal holds (1 -leading_coeff a) = a

    proof

      let a be Ordinal;

       not 1 in 1;

      

      hence (1 -leading_coeff a) = (a div^ ( exp (1 qua Ordinal, 0 ))) by ORDINAL5:def 10

      .= (a div^ 1) by ORDINAL2: 43

      .= a by ORDINAL3: 71;

    end;

    theorem :: ORDINAL7:41

    for b be Ordinal holds (b -leading_coeff 0 ) = 0 by ORDINAL3: 70;

    theorem :: ORDINAL7:42

    

     Th55: for a,b be Ordinal st a in b holds (b -leading_coeff a) = a

    proof

      let a,b be Ordinal;

      assume

       A1: a in b;

      per cases ;

        suppose 0 in a;

        

        thus (b -leading_coeff a) = (a div^ ( exp (b, 0 ))) by A1, Th21

        .= (a div^ 1) by ORDINAL2: 43

        .= a by ORDINAL3: 71;

      end;

        suppose not 0 in a;

        then a = 0 or a in 0 by ORDINAL1: 14;

        hence thesis by ORDINAL3: 70;

      end;

    end;

    theorem :: ORDINAL7:43

    for b be Ordinal holds (b -leading_coeff 1) = 1

    proof

      let b be Ordinal;

      per cases by ORDINAL1: 14;

        suppose 1 in b;

        hence thesis by Th55;

      end;

        suppose 1 = b;

        hence thesis by Th53;

      end;

        suppose b in 1;

        then b = 0 by TARSKI:def 1, CARD_1: 49;

        hence thesis by Th52;

      end;

    end;

    theorem :: ORDINAL7:44

    

     Th57: for a,b,c be Ordinal st c in b holds (b -leading_coeff (c *^ ( exp (b,a)))) = c

    proof

      let a,b,c be Ordinal;

      assume

       A1: c in b;

      per cases ;

        suppose

         A2: 0 in c;

        

         A3: 0 in ( exp (b,a)) by A1, ORDINAL1: 14;

        

        thus (b -leading_coeff (c *^ ( exp (b,a)))) = ((c *^ ( exp (b,a))) div^ ( exp (b,a))) by A1, A2, ORDINAL5: 58

        .= (((c *^ ( exp (b,a))) +^ 0 ) div^ ( exp (b,a))) by ORDINAL2: 27

        .= c by A3, ORDINAL3: 66;

      end;

        suppose not 0 in c;

        then

         A4: c = 0 by ORDINAL1: 14;

        

        hence (b -leading_coeff (c *^ ( exp (b,a)))) = (b -leading_coeff 0 ) by ORDINAL2: 35

        .= c by A4, ORDINAL3: 70;

      end;

    end;

    theorem :: ORDINAL7:45

    for a,b be Ordinal st 1 in b holds (b -leading_coeff ( exp (b,a))) = 1

    proof

      let a,b be Ordinal;

      assume

       A1: 1 in b;

      

      thus (b -leading_coeff ( exp (b,a))) = (b -leading_coeff (1 *^ ( exp (b,a)))) by ORDINAL2: 39

      .= 1 by A1, Th57;

    end;

    registration

      let c be Cantor-component Ordinal;

      cluster ( omega -leading_coeff c) -> natural non empty;

      coherence

      proof

        consider b be Ordinal, n be Nat such that

         A1: 0 in ( Segm n) & c = (n *^ ( exp ( omega ,b))) by ORDINAL5:def 9;

        thus thesis by A1, Th57, ORDINAL1:def 12;

      end;

    end

    theorem :: ORDINAL7:46

    

     Th59: for c be Cantor-component Ordinal holds c = (( omega -leading_coeff c) *^ ( exp ( omega ,( omega -exponent c))))

    proof

      let c be Cantor-component Ordinal;

      consider b be Ordinal, n be Nat such that

       A1: 0 in ( Segm n) & c = (n *^ ( exp ( omega ,b))) by ORDINAL5:def 9;

      

       A2: ( omega -leading_coeff c) = n by A1, Th57, ORDINAL1:def 12;

       0 in n & n in omega by A1, ORDINAL1:def 12;

      hence thesis by A1, A2, ORDINAL5: 58;

    end;

    definition

      let A be Ordinal-Sequence, b be Ordinal;

      :: ORDINAL7:def3

      func b -leading_coeff A -> Ordinal-Sequence means

      : Def3: ( dom it ) = ( dom A) & for a be object st a in ( dom A) holds (it . a) = (b -leading_coeff (A . a));

      existence

      proof

        deffunc F( object) = (b -leading_coeff (A . $1));

        consider f be Function such that

         A1: ( dom f) = ( dom A) & for a be object st a in ( dom A) holds (f . a) = F(a) from FUNCT_1:sch 3;

        reconsider f as Sequence by A1, ORDINAL1: 31;

        now

          reconsider c = ( sup ( rng f)) as Ordinal;

          take c;

          now

            let y be object;

            assume

             A2: y in ( rng f);

            then

            consider x be object such that

             A3: x in ( dom f) & (f . x) = y by FUNCT_1:def 3;

            (f . x) = (b -leading_coeff (A . x)) by A1, A3;

            hence y in ( sup ( rng f)) by A2, A3, ORDINAL2: 19;

          end;

          hence ( rng f) c= ( sup ( rng f)) by TARSKI:def 3;

        end;

        then

        reconsider f as Ordinal-Sequence by ORDINAL2:def 4;

        take f;

        thus thesis by A1;

      end;

      uniqueness

      proof

        let f1,f2 be Ordinal-Sequence;

        assume that

         A4: ( dom f1) = ( dom A) and

         A5: for a be object st a in ( dom A) holds (f1 . a) = (b -leading_coeff (A . a)) and

         A6: ( dom f2) = ( dom A) and

         A7: for a be object st a in ( dom A) holds (f2 . a) = (b -leading_coeff (A . a));

        now

          let a be object;

          assume

           A8: a in ( dom f1);

          

          hence (f1 . a) = (b -leading_coeff (A . a)) by A4, A5

          .= (f2 . a) by A4, A7, A8;

        end;

        hence thesis by A4, A6, FUNCT_1: 2;

      end;

    end

    registration

      let A be empty Ordinal-Sequence, b be Ordinal;

      cluster (b -leading_coeff A) -> empty;

      coherence

      proof

        ( dom A) = ( dom (b -leading_coeff A)) by Def3;

        hence thesis;

      end;

    end

    registration

      let A be non empty Ordinal-Sequence, b be Ordinal;

      cluster (b -leading_coeff A) -> non empty;

      coherence

      proof

        ( dom A) = ( dom (b -leading_coeff A)) by Def3;

        hence thesis;

      end;

    end

    registration

      let A be finite Ordinal-Sequence, b be Ordinal;

      cluster (b -leading_coeff A) -> finite;

      coherence

      proof

        ( dom A) = ( dom (b -leading_coeff A)) by Def3;

        hence thesis by FINSET_1: 10;

      end;

    end

    registration

      let A be infinite Ordinal-Sequence, b be Ordinal;

      cluster (b -leading_coeff A) -> infinite;

      coherence

      proof

        ( dom A) = ( dom (b -leading_coeff A)) by Def3;

        hence thesis by FINSET_1: 10;

      end;

    end

    theorem :: ORDINAL7:47

    

     Th60: for a,b be Ordinal holds (b -leading_coeff <%a%>) = <%(b -leading_coeff a)%>

    proof

      let a,b be Ordinal;

      

       A1: ( dom (b -leading_coeff <%a%>)) = ( dom <%a%>) by Def3

      .= 1 by AFINSQ_1:def 4;

       0 in 1 by TARSKI:def 1, CARD_1: 49;

      then 0 in ( dom <%a%>) by AFINSQ_1:def 4;

      

      then ((b -leading_coeff <%a%>) . 0 ) = (b -leading_coeff ( <%a%> . 0 )) by Def3

      .= (b -leading_coeff a);

      hence thesis by A1, AFINSQ_1:def 4;

    end;

    theorem :: ORDINAL7:48

    for A,B be Ordinal-Sequence, b be Ordinal holds (b -leading_coeff (A ^ B)) = ((b -leading_coeff A) ^ (b -leading_coeff B))

    proof

      let A,B be Ordinal-Sequence, b be Ordinal;

      

       A1: ( dom (b -leading_coeff (A ^ B))) = ( dom (A ^ B)) by Def3

      .= (( dom A) +^ ( dom B)) by ORDINAL4:def 1

      .= (( dom A) +^ ( dom (b -leading_coeff B))) by Def3

      .= (( dom (b -leading_coeff A)) +^ ( dom (b -leading_coeff B))) by Def3

      .= ( dom ((b -leading_coeff A) ^ (b -leading_coeff B))) by ORDINAL4:def 1;

      now

        let x be object;

        assume x in ( dom (b -leading_coeff (A ^ B)));

        then

         A2: x in ( dom (A ^ B)) by Def3;

        then

         A3: ((b -leading_coeff (A ^ B)) . x) = (b -leading_coeff ((A ^ B) . x)) by Def3;

        reconsider c = x as Ordinal by A2;

        c in ( dom A) or (( dom A) c= c & (c -^ ( dom A)) in ( dom B))

        proof

          assume not c in ( dom A);

          hence

           A4: ( dom A) c= c by ORDINAL1: 16;

          c in (( dom A) +^ ( dom B)) by A2, ORDINAL4:def 1;

          then (c -^ ( dom A)) in ((( dom A) +^ ( dom B)) -^ ( dom A)) by A4, ORDINAL3: 53;

          hence thesis by ORDINAL3: 52;

        end;

        per cases ;

          suppose

           A5: c in ( dom A);

          then

           A6: c in ( dom (b -leading_coeff A)) by Def3;

          ((A ^ B) . x) = (A . x) by A5, ORDINAL4:def 1;

          

          hence ((b -leading_coeff (A ^ B)) . x) = ((b -leading_coeff A) . x) by A3, A5, Def3

          .= (((b -leading_coeff A) ^ (b -leading_coeff B)) . x) by A6, ORDINAL4:def 1;

        end;

          suppose

           A7: ( dom A) c= c & (c -^ ( dom A)) in ( dom B);

          then

           A8: (c -^ ( dom A)) in ( dom (b -leading_coeff B)) by Def3;

          ((A ^ B) . x) = ((A ^ B) . (( dom A) +^ (c -^ ( dom A)))) by A7, ORDINAL3:def 5

          .= (B . (c -^ ( dom A))) by A7, ORDINAL4:def 1;

          

          hence ((b -leading_coeff (A ^ B)) . x) = ((b -leading_coeff B) . (c -^ ( dom A))) by A3, A7, Def3

          .= (((b -leading_coeff A) ^ (b -leading_coeff B)) . (( dom (b -leading_coeff A)) +^ (c -^ ( dom A)))) by A8, ORDINAL4:def 1

          .= (((b -leading_coeff A) ^ (b -leading_coeff B)) . (( dom A) +^ (c -^ ( dom A)))) by Def3

          .= (((b -leading_coeff A) ^ (b -leading_coeff B)) . x) by A7, ORDINAL3:def 5;

        end;

      end;

      hence thesis by A1, FUNCT_1: 2;

    end;

    theorem :: ORDINAL7:49

    for A be Ordinal-Sequence, b,c be Ordinal holds (b -leading_coeff (A | c)) = ((b -leading_coeff A) | c)

    proof

      let A be Ordinal-Sequence, b,c be Ordinal;

      

       A1: ( dom (b -leading_coeff (A | c))) = ( dom (A | c)) by Def3

      .= (( dom A) /\ c) by RELAT_1: 61

      .= (( dom (b -leading_coeff A)) /\ c) by Def3

      .= ( dom ((b -leading_coeff A) | c)) by RELAT_1: 61;

      now

        let x be object;

        assume

         A2: x in ( dom (b -leading_coeff (A | c)));

        then

         A3: x in ( dom (A | c)) by Def3;

        then

         A4: x in ( dom A) by RELAT_1: 57;

        

        thus ((b -leading_coeff (A | c)) . x) = (b -leading_coeff ((A | c) . x)) by A3, Def3

        .= (b -leading_coeff (A . x)) by A3, FUNCT_1: 47

        .= ((b -leading_coeff A) . x) by A4, Def3

        .= (((b -leading_coeff A) | c) . x) by A1, A2, FUNCT_1: 47;

      end;

      hence thesis by A1, FUNCT_1: 2;

    end;

    theorem :: ORDINAL7:50

    for A be finite Ordinal-Sequence, b be Ordinal, n be Nat holds (b -leading_coeff (A /^ n)) = ((b -leading_coeff A) /^ n)

    proof

      let A be finite Ordinal-Sequence, b be Ordinal, n be Nat;

      

       A1: ( dom (b -leading_coeff (A /^ n))) = ( len (A /^ n)) by Def3

      .= (( len A) -' n) by AFINSQ_2:def 2

      .= (( len (b -leading_coeff A)) -' n) by Def3

      .= ( dom ((b -leading_coeff A) /^ n)) by AFINSQ_2:def 2;

      now

        let k be Nat;

        assume

         A2: k in ( dom (b -leading_coeff (A /^ n)));

        then

         A3: k in ( dom (A /^ n)) by Def3;

        

         A4: (b -leading_coeff (A . (k + n))) = ((b -leading_coeff A) . (k + n))

        proof

          per cases ;

            suppose (k + n) in ( dom A);

            hence thesis by Def3;

          end;

            suppose

             A5: not (k + n) in ( dom A);

            then (A . (k + n)) = {} by FUNCT_1:def 2;

            then

             A6: (b -leading_coeff (A . (k + n))) = {} by ORDINAL3: 70;

             not (k + n) in ( dom (b -leading_coeff A)) by A5, Def3;

            hence thesis by A6, FUNCT_1:def 2;

          end;

        end;

        

        thus ((b -leading_coeff (A /^ n)) . k) = (b -leading_coeff ((A /^ n) . k)) by A3, Def3

        .= (b -leading_coeff (A . (k + n))) by A3, AFINSQ_2:def 2

        .= (((b -leading_coeff A) /^ n) . k) by A1, A2, A4, AFINSQ_2:def 2;

      end;

      hence thesis by A1, AFINSQ_1: 8;

    end;

    registration

      let A be Cantor-normal-form Ordinal-Sequence, a be object;

      cluster (( omega -leading_coeff A) . a) -> natural;

      coherence

      proof

        per cases ;

          suppose

           A1: a in ( dom A);

          then

           A2: (( omega -leading_coeff A) . a) = ( omega -leading_coeff (A . a)) by Def3;

          (A . a) is Cantor-component by A1, ORDINAL5:def 11;

          hence thesis by A2;

        end;

          suppose not a in ( dom A);

          then not a in ( dom ( omega -leading_coeff A)) by Def3;

          hence thesis by FUNCT_1:def 2;

        end;

      end;

    end

    registration

      let A be Cantor-normal-form Ordinal-Sequence;

      cluster ( omega -leading_coeff A) -> natural-valued non-empty;

      coherence

      proof

        now

          let y be object;

          assume y in ( rng ( omega -leading_coeff A));

          then

          consider x be object such that

           A1: x in ( dom ( omega -leading_coeff A)) & (( omega -leading_coeff A) . x) = y by FUNCT_1:def 3;

          thus y in NAT by A1, ORDINAL1:def 12;

        end;

        hence ( omega -leading_coeff A) is natural-valued by TARSKI:def 3, VALUED_0:def 6;

        now

          let x be object;

          assume x in ( dom ( omega -leading_coeff A));

          then

           A2: x in ( dom A) by Def3;

          then

           A3: (A . x) is Cantor-component by ORDINAL5:def 11;

          (( omega -leading_coeff A) . x) = ( omega -leading_coeff (A . x)) by A2, Def3;

          hence (( omega -leading_coeff A) . x) is non empty by A3;

        end;

        hence thesis by FUNCT_1:def 9;

      end;

    end

    theorem :: ORDINAL7:51

    

     Th64: for A be Cantor-normal-form Ordinal-Sequence, a be object st a in ( dom A) holds (A . a) = (( omega -leading_coeff (A . a)) *^ ( exp ( omega ,( omega -exponent (A . a)))))

    proof

      let A be Cantor-normal-form Ordinal-Sequence, a be object;

      assume a in ( dom A);

      then (A . a) is Cantor-component by ORDINAL5:def 11;

      hence thesis by Th59;

    end;

    theorem :: ORDINAL7:52

    

     Th65: for A be Cantor-normal-form Ordinal-Sequence, a be object st a in ( dom A) holds (A . a) = ((( omega -leading_coeff A) . a) *^ ( exp ( omega ,(( omega -exponent A) . a))))

    proof

      let A be Cantor-normal-form Ordinal-Sequence, a be object;

      assume

       A1: a in ( dom A);

      

      hence (A . a) = (( omega -leading_coeff (A . a)) *^ ( exp ( omega ,( omega -exponent (A . a))))) by Th64

      .= ((( omega -leading_coeff A) . a) *^ ( exp ( omega ,( omega -exponent (A . a))))) by A1, Def3

      .= ((( omega -leading_coeff A) . a) *^ ( exp ( omega ,(( omega -exponent A) . a)))) by A1, Def1;

    end;

    theorem :: ORDINAL7:53

    

     Th66: for A be decreasing Ordinal-Sequence holds for B be natural-valued non-empty Ordinal-Sequence st ( dom A) = ( dom B) holds ex C be Cantor-normal-form Ordinal-Sequence st ( omega -exponent C) = A & ( omega -leading_coeff C) = B

    proof

      let A be decreasing Ordinal-Sequence;

      let B be natural-valued non-empty Ordinal-Sequence;

      assume

       A1: ( dom A) = ( dom B);

      deffunc F( Ordinal) = ((B . $1) *^ ( exp ( omega ,(A . $1))));

      consider C be Ordinal-Sequence such that

       A2: ( dom C) = ( dom A) & for a be Ordinal st a in ( dom A) holds (C . a) = F(a) from ORDINAL2:sch 3;

       A3:

      now

        let a be Ordinal;

        assume

         A4: a in ( dom C);

        then

         A5: (C . a) = ((B . a) *^ ( exp ( omega ,(A . a)))) by A2;

        (B . a) <> {} by A1, A2, A4, FUNCT_1:def 9;

        hence (C . a) is Cantor-component by A5;

      end;

      now

        let a,b be Ordinal;

        assume

         A6: a in b & b in ( dom C);

        then

         A7: (C . a) = ((B . a) *^ ( exp ( omega ,(A . a)))) & (C . b) = ((B . b) *^ ( exp ( omega ,(A . b)))) by A2, ORDINAL1: 10;

        

         XA: ( rng B) c= NAT by VALUED_0:def 6;

        

         X0: b in ( dom B) by A6, A1, A2;

        then b c= ( dom B) by ORDINAL1:def 2;

        then

         xy: (B . b) in ( rng B) & (B . a) in ( rng B) by A6, X0, FUNCT_1: 3;

        (B . a) <> {} & (B . b) <> {} by A1, A2, A6, ORDINAL1: 10, FUNCT_1:def 9;

        then 0 c< (B . a) & 0 c< (B . b) by XBOOLE_1: 2, XBOOLE_0:def 8;

        then 0 in (B . a) & 0 in (B . b) by ORDINAL1: 11;

        then ( omega -exponent (C . b)) = (A . b) & ( omega -exponent (C . a)) = (A . a) by A7, ORDINAL5: 58, XA, xy;

        hence ( omega -exponent (C . b)) in ( omega -exponent (C . a)) by A2, A6, ORDINAL5:def 1;

      end;

      then

      reconsider C as Cantor-normal-form Ordinal-Sequence by A3, ORDINAL5:def 11;

      take C;

      

       A9: ( dom ( omega -exponent C)) = ( dom A) by A2, Def1;

      now

        let a be object;

        assume a in ( dom ( omega -exponent C));

        then

         A10: a in ( dom C) by Def1;

        then

         A11: (C . a) = ((B . a) *^ ( exp ( omega ,(A . a)))) by A2;

        (B . a) <> {} by A1, A2, A10, FUNCT_1:def 9;

        then 0 c< (B . a) by XBOOLE_1: 2, XBOOLE_0:def 8;

        then

         A12: 0 in (B . a) by ORDINAL1: 11;

        

         Sa: (B . a) in ( rng B) by FUNCT_1: 3, A10, A1, A2;

        ( rng B) c= NAT by VALUED_0:def 6;

        then ( omega -exponent (C . a)) = (A . a) by A11, A12, ORDINAL5: 58, Sa;

        hence (( omega -exponent C) . a) = (A . a) by A10, Def1;

      end;

      hence ( omega -exponent C) = A by A9, FUNCT_1: 2;

      

       A13: ( dom ( omega -leading_coeff C)) = ( dom B) by A1, A2, Def3;

      now

        let a be object;

        assume a in ( dom ( omega -leading_coeff C));

        then

         A14: a in ( dom C) by Def3;

        then (C . a) = ((B . a) *^ ( exp ( omega ,(A . a)))) by A2;

        then ( omega -leading_coeff (C . a)) = (B . a) by Th57, ORDINAL1:def 12;

        hence (( omega -leading_coeff C) . a) = (B . a) by A14, Def3;

      end;

      hence ( omega -leading_coeff C) = B by A13, FUNCT_1: 2;

    end;

    theorem :: ORDINAL7:54

    

     Th67: for A,B be Cantor-normal-form Ordinal-Sequence st ( omega -exponent A) = ( omega -exponent B) & ( omega -leading_coeff A) = ( omega -leading_coeff B) holds A = B

    proof

      let A,B be Cantor-normal-form Ordinal-Sequence;

      assume that

       A1: ( omega -exponent A) = ( omega -exponent B) and

       A2: ( omega -leading_coeff A) = ( omega -leading_coeff B);

      

       A3: ( dom A) = ( dom ( omega -exponent A)) by Def1

      .= ( dom B) by A1, Def1;

      now

        let a be object;

        assume

         A4: a in ( dom A);

        

        hence (A . a) = ((( omega -leading_coeff A) . a) *^ ( exp ( omega ,(( omega -exponent A) . a)))) by Th65

        .= (B . a) by A1, A2, A3, A4, Th65;

      end;

      hence thesis by A3, FUNCT_1: 2;

    end;

    definition

      let a be Ordinal;

      :: ORDINAL7:def4

      func CantorNF a -> Cantor-normal-form Ordinal-Sequence means

      : Def4: ( Sum^ it ) = a;

      existence by ORDINAL5: 69;

      uniqueness by Th45;

    end

    registration

      let a be Ordinal;

      reduce ( Sum^ ( CantorNF a)) to a;

      correctness by Def4;

    end

    registration

      let A be Cantor-normal-form Ordinal-Sequence;

      reduce ( CantorNF ( Sum^ A)) to A;

      correctness by Def4;

    end

    theorem :: ORDINAL7:55

    ( CantorNF {} ) = {} by ORDINAL5: 52;

    registration

      let a be empty Ordinal;

      cluster ( CantorNF a) -> empty;

      coherence by ORDINAL5: 52;

    end

    registration

      let a be non empty Ordinal;

      cluster ( CantorNF a) -> non empty;

      coherence by ORDINAL5: 52;

    end

    theorem :: ORDINAL7:56

    

     Th69: for a be Ordinal, n be non zero Nat holds ( CantorNF (n *^ ( exp ( omega ,a)))) = <%(n *^ ( exp ( omega ,a)))%>

    proof

      let a be Ordinal, n be non zero Nat;

      ( Sum^ <%(n *^ ( exp ( omega ,a)))%>) = (n *^ ( exp ( omega ,a))) by ORDINAL5: 53;

      hence thesis;

    end;

    theorem :: ORDINAL7:57

    

     Th70: for a be Cantor-component Ordinal holds ( CantorNF a) = <%a%>

    proof

      let a be Cantor-component Ordinal;

      ( Sum^ <%a%>) = a by ORDINAL5: 53;

      hence thesis;

    end;

    theorem :: ORDINAL7:58

    

     Th71: for n be non zero Nat holds ( CantorNF n) = <%n%>

    proof

      let n be non zero Nat;

      ( Sum^ <%n%>) = n by ORDINAL5: 53;

      hence thesis;

    end;

    theorem :: ORDINAL7:59

    for a be non empty Ordinal, n,m be non zero Nat holds ( CantorNF ((n *^ ( exp ( omega ,a))) +^ m)) = <%(n *^ ( exp ( omega ,a))), m%>

    proof

      let a be non empty Ordinal, n,m be non zero Nat;

      ( Sum^ <%(n *^ ( exp ( omega ,a))), m%>) = ((n *^ ( exp ( omega ,a))) +^ m) by Th25;

      hence thesis;

    end;

    theorem :: ORDINAL7:60

    

     Th73: for a be non empty Ordinal, b be Ordinal, n be non zero Nat st b in ( omega -exponent ( last ( CantorNF a))) holds ( CantorNF (a +^ (n *^ ( exp ( omega ,b))))) = (( CantorNF a) ^ <%(n *^ ( exp ( omega ,b)))%>)

    proof

      let a be non empty Ordinal, b be Ordinal, n be non zero Nat;

      assume

       A1: b in ( omega -exponent ( last ( CantorNF a)));

      set A = ( CantorNF a), B = <%(n *^ ( exp ( omega ,b)))%>;

      

       A2: (( CantorNF a) ^ <%(n *^ ( exp ( omega ,b)))%>) is Cantor-normal-form by A1, Th37;

      ( Sum^ (A ^ B)) = (( Sum^ A) +^ (n *^ ( exp ( omega ,b)))) by ORDINAL5: 54

      .= (a +^ (n *^ ( exp ( omega ,b))));

      hence thesis by A2;

    end;

    theorem :: ORDINAL7:61

    for a be non empty Ordinal, n be non zero Nat st ( omega -exponent ( last ( CantorNF a))) <> 0 holds ( CantorNF (a +^ n)) = (( CantorNF a) ^ <%n%>)

    proof

      let a be non empty Ordinal, n be non zero Nat;

      assume ( omega -exponent ( last ( CantorNF a))) <> 0 ;

      then 0 c< ( omega -exponent ( last ( CantorNF a))) by XBOOLE_1: 2, XBOOLE_0:def 8;

      then

       A1: 0 in ( omega -exponent ( last ( CantorNF a))) by ORDINAL1: 11;

      

      thus ( CantorNF (a +^ n)) = ( CantorNF (a +^ (n *^ 1))) by ORDINAL2: 39

      .= ( CantorNF (a +^ (n *^ ( exp ( omega , 0 qua Ordinal))))) by ORDINAL2: 43

      .= (( CantorNF a) ^ <%(n *^ ( exp ( omega , 0 qua Ordinal)))%>) by A1, Th73

      .= (( CantorNF a) ^ <%(n *^ 1)%>) by ORDINAL2: 43

      .= (( CantorNF a) ^ <%n%>) by ORDINAL2: 39;

    end;

    theorem :: ORDINAL7:62

    for a be non empty Ordinal, b be Ordinal, n be non zero Nat st ( omega -exponent (( CantorNF a) . 0 )) in b holds ( CantorNF ((n *^ ( exp ( omega ,b))) +^ a)) = ( <%(n *^ ( exp ( omega ,b)))%> ^ ( CantorNF a))

    proof

      let a be non empty Ordinal, b be Ordinal, n be non zero Nat;

      assume ( omega -exponent (( CantorNF a) . 0 )) in b;

      then

       A1: ( <%(n *^ ( exp ( omega ,b)))%> ^ ( CantorNF a)) is Cantor-normal-form by Th39;

      set A = <%(n *^ ( exp ( omega ,b)))%>, B = ( CantorNF a);

      ( Sum^ (A ^ B)) = ((n *^ ( exp ( omega ,b))) +^ ( Sum^ B)) by ORDINAL5: 55

      .= ((n *^ ( exp ( omega ,b))) +^ a);

      hence thesis by A1;

    end;

    begin

    definition

      let a,b be Ordinal;

      :: ORDINAL7:def5

      func a (+) b -> Ordinal means

      : Def5: ex C be Cantor-normal-form Ordinal-Sequence st it = ( Sum^ C) & ( rng ( omega -exponent C)) = (( rng ( omega -exponent ( CantorNF a))) \/ ( rng ( omega -exponent ( CantorNF b)))) & for d be object st d in ( dom C) holds (( omega -exponent (C . d)) in (( rng ( omega -exponent ( CantorNF a))) \ ( rng ( omega -exponent ( CantorNF b)))) implies ( omega -leading_coeff (C . d)) = (( omega -leading_coeff ( CantorNF a)) . ((( omega -exponent ( CantorNF a)) " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng ( omega -exponent ( CantorNF b))) \ ( rng ( omega -exponent ( CantorNF a)))) implies ( omega -leading_coeff (C . d)) = (( omega -leading_coeff ( CantorNF b)) . ((( omega -exponent ( CantorNF b)) " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng ( omega -exponent ( CantorNF a))) /\ ( rng ( omega -exponent ( CantorNF b)))) implies ( omega -leading_coeff (C . d)) = ((( omega -leading_coeff ( CantorNF a)) . ((( omega -exponent ( CantorNF a)) " ) . ( omega -exponent (C . d)))) + (( omega -leading_coeff ( CantorNF b)) . ((( omega -exponent ( CantorNF b)) " ) . ( omega -exponent (C . d))))));

      existence

      proof

        set R = (( rng ( omega -exponent ( CantorNF a))) \/ ( rng ( omega -exponent ( CantorNF b))));

        set c = ( sup R);

        set RS = RelStr (# c, ( RelIncl c) #);

        for x be object holds x in R implies x in the carrier of RS by ORDINAL2: 19;

        then

        reconsider R as finite Subset of RS by TARSKI:def 3;

        now

          let x,y be object;

          assume

           A1: x in R & y in R & x <> y;

          then

          reconsider A = x, B = y as Ordinal;

          A c= B or B c= A;

          hence [x, y] in the InternalRel of RS or [y, x] in the InternalRel of RS by A1, WELLORD2:def 1;

        end;

        then

        consider e0 be FinSequence of RS such that

         A2: e0 is R -desc_ordering by RELAT_2:def 6, ORDERS_5: 78;

        set e = ( FS2XFS e0);

        

         A3: ( rng e) = ( rng e0) by Th15

        .= R by A2, ORDERS_5:def 26;

        reconsider e as Ordinal-Sequence;

        now

          let a,b be Ordinal;

          assume

           A4: a in b & b in ( dom e);

          then

           A5: a in ( dom e) by ORDINAL1: 10;

          ( dom e) in omega by CARD_1: 61;

          then a in omega & b in omega by A4, A5, ORDINAL1: 10;

          then

          reconsider n = a, m = b as Nat;

          ( card ( Segm n)) in ( card ( Segm m)) by A4;

          then

           A6: (n + 1) < (m + 1) by NAT_1: 41, XREAL_1: 8;

          

           A7: (n + 1) in ( dom e0) & (m + 1) in ( dom e0) by A4, A5, Th13;

          then (e0 /. (m + 1)) < (e0 /. (n + 1)) by A2, A6, ORDERS_5:def 22;

          then

           A8: [(e0 /. (m + 1)), (e0 /. (n + 1))] in the InternalRel of RS & (e0 /. (m + 1)) <> (e0 /. (n + 1)) by ORDERS_2:def 5, ORDERS_2:def 6;

          

           A9: (e0 /. (m + 1)) = (e0 . (m + 1)) & (e0 /. (n + 1)) = (e0 . (n + 1)) by A7, PARTFUN1:def 6;

          (e0 . (m + 1)) in ( rng e0) & (e0 . (n + 1)) in ( rng e0) by A7, FUNCT_1: 3;

          then (e0 . (m + 1)) in R & (e0 . (n + 1)) in R by A2, ORDERS_5:def 26;

          then (e0 . (m + 1)) c= (e0 . (n + 1)) by A8, A9, WELLORD2:def 1;

          then

           A10: (e0 . (m + 1)) c< (e0 . (n + 1)) by A8, A9, XBOOLE_0:def 8;

          (n + 1) <= ( len e0) & (m + 1) <= ( len e0) by A7, FINSEQ_3: 25;

          then ((n + 1) - 1) < (( len e0) - 0 ) & ((m + 1) - 1) < (( len e0) - 0 ) by XREAL_1: 15;

          then (e . n) = (e0 . (n + 1)) & (e . m) = (e0 . (m + 1)) by AFINSQ_1:def 8;

          hence (e . b) in (e . a) by A10, ORDINAL1: 11;

        end;

        then

        reconsider e as decreasing Ordinal-Sequence by ORDINAL5:def 1;

        set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

        set L1 = ( omega -leading_coeff ( CantorNF a));

        set L2 = ( omega -leading_coeff ( CantorNF b));

        defpred P[ object, object] means ((e . $1) in (( rng E1) \ ( rng E2)) implies $2 = (L1 . ((E1 " ) . (e . $1)))) & ((e . $1) in (( rng E2) \ ( rng E1)) implies $2 = (L2 . ((E2 " ) . (e . $1)))) & ((e . $1) in (( rng E1) /\ ( rng E2)) implies $2 = ((L1 . ((E1 " ) . (e . $1))) + (L2 . ((E2 " ) . (e . $1)))));

         A11:

        now

          let x,y1,y2 be object;

          assume

           A12: x in ( dom e) & P[x, y1] & P[x, y2];

          then (e . x) in R by A3, FUNCT_1: 3;

          per cases by XBOOLE_0:def 3;

            suppose (e . x) in ( rng E1) & not (e . x) in ( rng E2);

            hence y1 = y2 by A12, XBOOLE_0:def 5;

          end;

            suppose (e . x) in ( rng E2) & not (e . x) in ( rng E1);

            hence y1 = y2 by A12, XBOOLE_0:def 5;

          end;

            suppose (e . x) in ( rng E1) & (e . x) in ( rng E2);

            hence y1 = y2 by A12, XBOOLE_0:def 4;

          end;

        end;

        

         A13: for x be object st x in ( dom e) holds ex y be object st P[x, y]

        proof

          let x be object;

          assume x in ( dom e);

          then (e . x) in R by A3, FUNCT_1: 3;

          per cases by XBOOLE_0:def 3;

            suppose

             A14: (e . x) in ( rng E1) & not (e . x) in ( rng E2);

            take (L1 . ((E1 " ) . (e . x)));

            thus thesis by A14, XBOOLE_0:def 4;

          end;

            suppose

             A15: (e . x) in ( rng E2) & not (e . x) in ( rng E1);

            take (L2 . ((E2 " ) . (e . x)));

            thus thesis by A15, XBOOLE_0:def 4;

          end;

            suppose

             A16: (e . x) in ( rng E1) & (e . x) in ( rng E2);

            take ((L1 . ((E1 " ) . (e . x))) + (L2 . ((E2 " ) . (e . x))));

            thus thesis by A16, XBOOLE_0:def 5;

          end;

        end;

        consider f be Function such that

         A17: ( dom f) = ( dom e) and

         A18: for x be object st x in ( dom e) holds P[x, (f . x)] from FUNCT_1:sch 2( A11, A13);

        reconsider f as Sequence by A17, ORDINAL1: 31;

        now

          let y be object;

          assume y in ( rng f);

          then

          consider x be object such that

           A19: x in ( dom f) & (f . x) = y by FUNCT_1:def 3;

          (e . x) in ( rng e) by A17, A19, FUNCT_1: 3;

          per cases by A3, XBOOLE_0:def 3;

            suppose (e . x) in ( rng E1) & not (e . x) in ( rng E2);

            then (e . x) in (( rng E1) \ ( rng E2)) by XBOOLE_0:def 5;

            then (f . x) = (L1 . ((E1 " ) . (e . x))) by A17, A18, A19;

            hence y in NAT by A19, ORDINAL1:def 12;

          end;

            suppose (e . x) in ( rng E2) & not (e . x) in ( rng E1);

            then (e . x) in (( rng E2) \ ( rng E1)) by XBOOLE_0:def 5;

            then (f . x) = (L2 . ((E2 " ) . (e . x))) by A17, A18, A19;

            hence y in NAT by A19, ORDINAL1:def 12;

          end;

            suppose (e . x) in ( rng E1) & (e . x) in ( rng E2);

            then (e . x) in (( rng E1) /\ ( rng E2)) by XBOOLE_0:def 4;

            then (f . x) = ((L1 . ((E1 " ) . (e . x))) + (L2 . ((E2 " ) . (e . x)))) by A17, A18, A19;

            hence y in NAT by A19, ORDINAL1:def 12;

          end;

        end;

        then f is natural-valued by TARSKI:def 3, VALUED_0:def 6;

        then

        reconsider f as natural-valued Ordinal-Sequence;

        now

          let x be object;

          assume

           A20: x in ( dom f);

          

           A21: (e . x) in ( rng E1) implies ((E1 " ) . (e . x)) in ( dom L1)

          proof

            assume (e . x) in ( rng E1);

            then (e . x) in ( dom (E1 " )) by FUNCT_1: 33;

            then ((E1 " ) . (e . x)) in ( rng (E1 " )) by FUNCT_1: 3;

            then ((E1 " ) . (e . x)) in ( dom E1) by FUNCT_1: 33;

            then ((E1 " ) . (e . x)) in ( dom ( CantorNF a)) by Def1;

            hence ((E1 " ) . (e . x)) in ( dom L1) by Def3;

          end;

          

           A22: (e . x) in ( rng E2) implies ((E2 " ) . (e . x)) in ( dom L2)

          proof

            assume (e . x) in ( rng E2);

            then (e . x) in ( dom (E2 " )) by FUNCT_1: 33;

            then ((E2 " ) . (e . x)) in ( rng (E2 " )) by FUNCT_1: 3;

            then ((E2 " ) . (e . x)) in ( dom E2) by FUNCT_1: 33;

            then ((E2 " ) . (e . x)) in ( dom ( CantorNF b)) by Def1;

            hence ((E2 " ) . (e . x)) in ( dom L2) by Def3;

          end;

          (e . x) in ( rng e) by A17, A20, FUNCT_1: 3;

          per cases by A3, XBOOLE_0:def 3;

            suppose

             A23: (e . x) in ( rng E1) & not (e . x) in ( rng E2);

            then (e . x) in (( rng E1) \ ( rng E2)) by XBOOLE_0:def 5;

            then (f . x) = (L1 . ((E1 " ) . (e . x))) by A17, A18, A20;

            hence (f . x) is non empty by A21, A23, FUNCT_1:def 9;

          end;

            suppose

             A24: (e . x) in ( rng E2) & not (e . x) in ( rng E1);

            then (e . x) in (( rng E2) \ ( rng E1)) by XBOOLE_0:def 5;

            then (f . x) = (L2 . ((E2 " ) . (e . x))) by A17, A18, A20;

            hence (f . x) is non empty by A22, A24, FUNCT_1:def 9;

          end;

            suppose

             A25: (e . x) in ( rng E1) & (e . x) in ( rng E2);

            then (e . x) in (( rng E1) /\ ( rng E2)) by XBOOLE_0:def 4;

            then

             A26: (f . x) = ((L1 . ((E1 " ) . (e . x))) + (L2 . ((E2 " ) . (e . x)))) by A17, A18, A20;

            (L1 . ((E1 " ) . (e . x))) <> {} & (L2 . ((E2 " ) . (e . x))) <> {} by A21, A22, A25, FUNCT_1:def 9;

            hence (f . x) is non empty by A26;

          end;

        end;

        then

        reconsider f as natural-valued non-empty Ordinal-Sequence by FUNCT_1:def 9;

        consider C be Cantor-normal-form Ordinal-Sequence such that

         A27: ( omega -exponent C) = e & ( omega -leading_coeff C) = f by A17, Th66;

        take ( Sum^ C), C;

        thus ( Sum^ C) = ( Sum^ C);

        thus ( rng ( omega -exponent C)) = (( rng E1) \/ ( rng E2)) by A3, A27;

        let d be object;

        assume

         A28: d in ( dom C);

        hereby

          assume ( omega -exponent (C . d)) in (( rng E1) \ ( rng E2));

          then

           A29: (e . d) in (( rng E1) \ ( rng E2)) by A27, A28, Def1;

          d in ( dom ( omega -exponent C)) by A28, Def1;

          

          then (f . d) = (L1 . ((E1 " ) . (e . d))) by A18, A27, A29

          .= (L1 . ((E1 " ) . ( omega -exponent (C . d)))) by A27, A28, Def1;

          hence ( omega -leading_coeff (C . d)) = (L1 . ((E1 " ) . ( omega -exponent (C . d)))) by A27, A28, Def3;

        end;

        hereby

          assume ( omega -exponent (C . d)) in (( rng E2) \ ( rng E1));

          then

           A30: (e . d) in (( rng E2) \ ( rng E1)) by A27, A28, Def1;

          d in ( dom ( omega -exponent C)) by A28, Def1;

          

          then (f . d) = (L2 . ((E2 " ) . (e . d))) by A18, A27, A30

          .= (L2 . ((E2 " ) . ( omega -exponent (C . d)))) by A27, A28, Def1;

          hence ( omega -leading_coeff (C . d)) = (L2 . ((E2 " ) . ( omega -exponent (C . d)))) by A27, A28, Def3;

        end;

        assume ( omega -exponent (C . d)) in (( rng E1) /\ ( rng E2));

        then

         A31: (e . d) in (( rng E1) /\ ( rng E2)) by A27, A28, Def1;

        d in ( dom ( omega -exponent C)) by A28, Def1;

        

        then (f . d) = ((L1 . ((E1 " ) . (e . d))) + (L2 . ((E2 " ) . (e . d)))) by A18, A27, A31

        .= ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . (e . d)))) by A27, A28, Def1

        .= ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d))))) by A27, A28, Def1;

        hence ( omega -leading_coeff (C . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d))))) by A27, A28, Def3;

      end;

      uniqueness

      proof

        let s1,s2 be Ordinal;

        set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

        set L1 = ( omega -leading_coeff ( CantorNF a));

        set L2 = ( omega -leading_coeff ( CantorNF b));

        assume that

         A32: ex C be Cantor-normal-form Ordinal-Sequence st s1 = ( Sum^ C) & ( rng ( omega -exponent C)) = (( rng E1) \/ ( rng E2)) & for d be object st d in ( dom C) holds (( omega -exponent (C . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C . d)) = (L1 . ((E1 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C . d)) = (L2 . ((E2 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d)))))) and

         A33: ex C be Cantor-normal-form Ordinal-Sequence st s2 = ( Sum^ C) & ( rng ( omega -exponent C)) = (( rng E1) \/ ( rng E2)) & for d be object st d in ( dom C) holds (( omega -exponent (C . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C . d)) = (L1 . ((E1 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C . d)) = (L2 . ((E2 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d))))));

        consider C1 be Cantor-normal-form Ordinal-Sequence such that

         A34: s1 = ( Sum^ C1) and

         A35: ( rng ( omega -exponent C1)) = (( rng E1) \/ ( rng E2)) and

         A36: for d be object st d in ( dom C1) holds (( omega -exponent (C1 . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C1 . d)) = (L1 . ((E1 " ) . ( omega -exponent (C1 . d))))) & (( omega -exponent (C1 . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C1 . d)) = (L2 . ((E2 " ) . ( omega -exponent (C1 . d))))) & (( omega -exponent (C1 . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C1 . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C1 . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C1 . d)))))) by A32;

        consider C2 be Cantor-normal-form Ordinal-Sequence such that

         A37: s2 = ( Sum^ C2) and

         A38: ( rng ( omega -exponent C2)) = (( rng E1) \/ ( rng E2)) and

         A39: for d be object st d in ( dom C2) holds (( omega -exponent (C2 . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C2 . d)) = (L1 . ((E1 " ) . ( omega -exponent (C2 . d))))) & (( omega -exponent (C2 . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C2 . d)) = (L2 . ((E2 " ) . ( omega -exponent (C2 . d))))) & (( omega -exponent (C2 . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C2 . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C2 . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C2 . d)))))) by A33;

        

         A40: ( dom C1) = ( card ( dom ( omega -exponent C1))) by Def1

        .= ( card ( rng ( omega -exponent C2))) by A35, A38, CARD_1: 70

        .= ( card ( dom ( omega -exponent C2))) by CARD_1: 70

        .= ( dom C2) by Def1;

        for x be object st x in ( dom C1) holds (C1 . x) = (C2 . x)

        proof

          let x be object;

          set e1 = ( omega -exponent (C1 . x)), e2 = ( omega -exponent (C2 . x));

          assume

           A41: x in ( dom C1);

          

          then

           A42: e1 = (( omega -exponent C1) . x) by Def1

          .= (( omega -exponent C2) . x) by A35, A38, Th34

          .= e2 by A40, A41, Def1;

          x in ( dom ( omega -exponent C1)) by A41, Def1;

          then (( omega -exponent C1) . x) in ( rng ( omega -exponent C1)) by FUNCT_1: 3;

          then

           A43: e1 in (( rng E1) \/ ( rng E2)) by A35, A41, Def1;

          

           A44: ( omega -leading_coeff (C1 . x)) = ( omega -leading_coeff (C2 . x))

          proof

            per cases by A43, XBOOLE_0:def 3;

              suppose e1 in ( rng E1) & not e1 in ( rng E2);

              then

               A45: e1 in (( rng E1) \ ( rng E2)) by XBOOLE_0:def 5;

              

              hence ( omega -leading_coeff (C1 . x)) = (L1 . ((E1 " ) . e2)) by A36, A41, A42

              .= ( omega -leading_coeff (C2 . x)) by A39, A40, A41, A42, A45;

            end;

              suppose e1 in ( rng E2) & not e1 in ( rng E1);

              then

               A46: e1 in (( rng E2) \ ( rng E1)) by XBOOLE_0:def 5;

              

              hence ( omega -leading_coeff (C1 . x)) = (L2 . ((E2 " ) . e2)) by A36, A41, A42

              .= ( omega -leading_coeff (C2 . x)) by A39, A40, A41, A42, A46;

            end;

              suppose e1 in ( rng E1) & e1 in ( rng E2);

              then

               A47: e1 in (( rng E1) /\ ( rng E2)) by XBOOLE_0:def 4;

              

              hence ( omega -leading_coeff (C1 . x)) = ((L1 . ((E1 " ) . e2)) + (L2 . ((E2 " ) . e2))) by A36, A41, A42

              .= ( omega -leading_coeff (C2 . x)) by A39, A40, A41, A42, A47;

            end;

          end;

          

          thus (C1 . x) = (( omega -leading_coeff (C1 . x)) *^ ( exp ( omega ,e1))) by A41, Th64

          .= (C2 . x) by A40, A41, A42, A44, Th64;

        end;

        hence s1 = s2 by A34, A37, A40, FUNCT_1: 2;

      end;

      commutativity

      proof

        let s,a,b be Ordinal;

        set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

        set L1 = ( omega -leading_coeff ( CantorNF a));

        set L2 = ( omega -leading_coeff ( CantorNF b));

        given C be Cantor-normal-form Ordinal-Sequence such that

         A48: s = ( Sum^ C) & ( rng ( omega -exponent C)) = (( rng E1) \/ ( rng E2)) and

         A49: for d be object st d in ( dom C) holds (( omega -exponent (C . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C . d)) = (L1 . ((E1 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C . d)) = (L2 . ((E2 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d))))));

        take C;

        thus s = ( Sum^ C) & ( rng ( omega -exponent C)) = (( rng E2) \/ ( rng E1)) by A48;

        thus thesis by A49;

      end;

    end

    theorem :: ORDINAL7:63

    

     Th76: for a,b be Ordinal holds ( rng ( omega -exponent ( CantorNF (a (+) b)))) = (( rng ( omega -exponent ( CantorNF a))) \/ ( rng ( omega -exponent ( CantorNF b))))

    proof

      let a,b be Ordinal;

      set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

      set L1 = ( omega -leading_coeff ( CantorNF a));

      set L2 = ( omega -leading_coeff ( CantorNF b));

      consider C be Cantor-normal-form Ordinal-Sequence such that

       A1: (a (+) b) = ( Sum^ C) & ( rng ( omega -exponent C)) = (( rng E1) \/ ( rng E2)) and for d be object st d in ( dom C) holds (( omega -exponent (C . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C . d)) = (L1 . ((E1 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C . d)) = (L2 . ((E2 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d)))))) by Def5;

      thus thesis by A1;

    end;

    theorem :: ORDINAL7:64

    

     Th77: for a,b be Ordinal holds ( dom ( CantorNF a)) c= ( dom ( CantorNF (a (+) b)))

    proof

      let a,b be Ordinal;

      set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

      set C0 = ( CantorNF (a (+) b));

      

       A1: ( dom ( CantorNF a)) = ( card ( dom E1)) by Def1

      .= ( card ( rng E1)) by CARD_1: 70;

      ( card ( rng E1)) c= ( card (( rng E1) \/ ( rng E2))) by XBOOLE_1: 7, CARD_1: 11;

      then ( card ( rng E1)) c= ( card ( rng ( omega -exponent C0))) by Th76;

      then ( dom ( CantorNF a)) c= ( card ( dom ( omega -exponent C0))) by A1, CARD_1: 70;

      hence ( dom ( CantorNF a)) c= ( dom C0) by Def1;

    end;

    theorem :: ORDINAL7:65

    

     Th78: for a,b be Ordinal, d be object st d in ( dom ( CantorNF (a (+) b))) & ( omega -exponent (( CantorNF (a (+) b)) . d)) in (( rng ( omega -exponent ( CantorNF a))) \ ( rng ( omega -exponent ( CantorNF b)))) holds ( omega -leading_coeff (( CantorNF (a (+) b)) . d)) = (( omega -leading_coeff ( CantorNF a)) . ((( omega -exponent ( CantorNF a)) " ) . ( omega -exponent (( CantorNF (a (+) b)) . d))))

    proof

      let a,b be Ordinal;

      set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

      set L1 = ( omega -leading_coeff ( CantorNF a));

      set L2 = ( omega -leading_coeff ( CantorNF b));

      consider C be Cantor-normal-form Ordinal-Sequence such that

       A1: (a (+) b) = ( Sum^ C) & ( rng ( omega -exponent C)) = (( rng E1) \/ ( rng E2)) and

       A2: for d be object st d in ( dom C) holds (( omega -exponent (C . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C . d)) = (L1 . ((E1 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C . d)) = (L2 . ((E2 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d)))))) by Def5;

      let d be object;

      assume d in ( dom ( CantorNF (a (+) b))) & ( omega -exponent (( CantorNF (a (+) b)) . d)) in (( rng ( omega -exponent ( CantorNF a))) \ ( rng ( omega -exponent ( CantorNF b))));

      hence thesis by A1, A2;

    end;

    theorem :: ORDINAL7:66

    

     Th79: for a,b be Ordinal, d be object st d in ( dom ( CantorNF (a (+) b))) & ( omega -exponent (( CantorNF (a (+) b)) . d)) in (( rng ( omega -exponent ( CantorNF b))) \ ( rng ( omega -exponent ( CantorNF a)))) holds ( omega -leading_coeff (( CantorNF (a (+) b)) . d)) = (( omega -leading_coeff ( CantorNF b)) . ((( omega -exponent ( CantorNF b)) " ) . ( omega -exponent (( CantorNF (a (+) b)) . d)))) by Th78;

    theorem :: ORDINAL7:67

    

     Th80: for a,b be Ordinal, d be object st d in ( dom ( CantorNF (a (+) b))) & ( omega -exponent (( CantorNF (a (+) b)) . d)) in (( rng ( omega -exponent ( CantorNF a))) /\ ( rng ( omega -exponent ( CantorNF b)))) holds ( omega -leading_coeff (( CantorNF (a (+) b)) . d)) = ((( omega -leading_coeff ( CantorNF a)) . ((( omega -exponent ( CantorNF a)) " ) . ( omega -exponent (( CantorNF (a (+) b)) . d)))) + (( omega -leading_coeff ( CantorNF b)) . ((( omega -exponent ( CantorNF b)) " ) . ( omega -exponent (( CantorNF (a (+) b)) . d)))))

    proof

      let a,b be Ordinal;

      set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

      set L1 = ( omega -leading_coeff ( CantorNF a));

      set L2 = ( omega -leading_coeff ( CantorNF b));

      consider C be Cantor-normal-form Ordinal-Sequence such that

       A1: (a (+) b) = ( Sum^ C) & ( rng ( omega -exponent C)) = (( rng E1) \/ ( rng E2)) and

       A2: for d be object st d in ( dom C) holds (( omega -exponent (C . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C . d)) = (L1 . ((E1 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C . d)) = (L2 . ((E2 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d)))))) by Def5;

      let d be object;

      assume d in ( dom ( CantorNF (a (+) b))) & ( omega -exponent (( CantorNF (a (+) b)) . d)) in (( rng ( omega -exponent ( CantorNF a))) /\ ( rng ( omega -exponent ( CantorNF b))));

      hence thesis by A1, A2;

    end;

    theorem :: ORDINAL7:68

    

     Th81: for a,b,c be Ordinal holds ((a (+) b) (+) c) = (a (+) (b (+) c))

    proof

      let a,b,c be Ordinal;

      set s4 = (a (+) b), s5 = (b (+) c), s6 = (s4 (+) c), s7 = (a (+) s5);

      set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

      set E3 = ( omega -exponent ( CantorNF c)), E4 = ( omega -exponent ( CantorNF s4));

      set E5 = ( omega -exponent ( CantorNF s5));

      set L1 = ( omega -leading_coeff ( CantorNF a));

      set L2 = ( omega -leading_coeff ( CantorNF b));

      set L3 = ( omega -leading_coeff ( CantorNF c));

      set L4 = ( omega -leading_coeff ( CantorNF s4));

      set L5 = ( omega -leading_coeff ( CantorNF s5));

      consider C4 be Cantor-normal-form Ordinal-Sequence such that

       A1: s4 = ( Sum^ C4) & ( rng ( omega -exponent C4)) = (( rng E1) \/ ( rng E2)) and

       A2: for d be object st d in ( dom C4) holds (( omega -exponent (C4 . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C4 . d)) = (L1 . ((E1 " ) . ( omega -exponent (C4 . d))))) & (( omega -exponent (C4 . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C4 . d)) = (L2 . ((E2 " ) . ( omega -exponent (C4 . d))))) & (( omega -exponent (C4 . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C4 . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C4 . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C4 . d)))))) by Def5;

      consider C5 be Cantor-normal-form Ordinal-Sequence such that

       A3: s5 = ( Sum^ C5) & ( rng ( omega -exponent C5)) = (( rng E2) \/ ( rng E3)) and

       A4: for d be object st d in ( dom C5) holds (( omega -exponent (C5 . d)) in (( rng E2) \ ( rng E3)) implies ( omega -leading_coeff (C5 . d)) = (L2 . ((E2 " ) . ( omega -exponent (C5 . d))))) & (( omega -exponent (C5 . d)) in (( rng E3) \ ( rng E2)) implies ( omega -leading_coeff (C5 . d)) = (L3 . ((E3 " ) . ( omega -exponent (C5 . d))))) & (( omega -exponent (C5 . d)) in (( rng E2) /\ ( rng E3)) implies ( omega -leading_coeff (C5 . d)) = ((L2 . ((E2 " ) . ( omega -exponent (C5 . d)))) + (L3 . ((E3 " ) . ( omega -exponent (C5 . d)))))) by Def5;

      consider C6 be Cantor-normal-form Ordinal-Sequence such that

       A5: s6 = ( Sum^ C6) & ( rng ( omega -exponent C6)) = (( rng E4) \/ ( rng E3)) and

       A6: for d be object st d in ( dom C6) holds (( omega -exponent (C6 . d)) in (( rng E4) \ ( rng E3)) implies ( omega -leading_coeff (C6 . d)) = (L4 . ((E4 " ) . ( omega -exponent (C6 . d))))) & (( omega -exponent (C6 . d)) in (( rng E3) \ ( rng E4)) implies ( omega -leading_coeff (C6 . d)) = (L3 . ((E3 " ) . ( omega -exponent (C6 . d))))) & (( omega -exponent (C6 . d)) in (( rng E4) /\ ( rng E3)) implies ( omega -leading_coeff (C6 . d)) = ((L4 . ((E4 " ) . ( omega -exponent (C6 . d)))) + (L3 . ((E3 " ) . ( omega -exponent (C6 . d)))))) by Def5;

      consider C7 be Cantor-normal-form Ordinal-Sequence such that

       A7: s7 = ( Sum^ C7) & ( rng ( omega -exponent C7)) = (( rng E1) \/ ( rng E5)) and

       A8: for d be object st d in ( dom C7) holds (( omega -exponent (C7 . d)) in (( rng E1) \ ( rng E5)) implies ( omega -leading_coeff (C7 . d)) = (L1 . ((E1 " ) . ( omega -exponent (C7 . d))))) & (( omega -exponent (C7 . d)) in (( rng E5) \ ( rng E1)) implies ( omega -leading_coeff (C7 . d)) = (L5 . ((E5 " ) . ( omega -exponent (C7 . d))))) & (( omega -exponent (C7 . d)) in (( rng E1) /\ ( rng E5)) implies ( omega -leading_coeff (C7 . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C7 . d)))) + (L5 . ((E5 " ) . ( omega -exponent (C7 . d)))))) by Def5;

      

       A9: ( rng E4) = (( rng E1) \/ ( rng E2)) by A1;

      

       A10: ( rng E5) = (( rng E2) \/ ( rng E3)) by A3;

      

       A11: ( omega -exponent C6) = ( omega -exponent C7) by A1, A3, A5, A7, Th34, XBOOLE_1: 4;

      

       A12: ( dom C6) = ( dom ( omega -exponent C6)) by Def1

      .= ( dom C7) by A11, Def1;

      for x be object st x in ( dom C6) holds (C6 . x) = (C7 . x)

      proof

        let x be object;

        assume

         A13: x in ( dom C6);

        then

         A14: x in ( dom ( omega -exponent C6)) & x in ( dom C7) by A12, Def1;

        

         A15: ( rng E1) c= ( rng E4) & ( rng E2) c= ( rng E4) & ( rng E2) c= ( rng E5) & ( rng E3) c= ( rng E5) by A1, A3, XBOOLE_1: 7;

        set e = ( omega -exponent (C6 . x));

        set x1 = ((E4 " ) . ( omega -exponent (C6 . x))), y1 = ((E3 " ) . ( omega -exponent (C6 . x)));

        set x2 = ((E1 " ) . ( omega -exponent (C7 . x))), y2 = ((E5 " ) . ( omega -exponent (C7 . x)));

        

         A16: e = (( omega -exponent C6) . x) by A13, Def1;

        then

         A17: e = ( omega -exponent (C7 . x)) by A11, A14, Def1;

        

         A18: ( omega -exponent (C6 . x)) in ( rng E4) implies x1 in ( dom C4) & ( omega -exponent (C4 . x1)) = ( omega -exponent (C6 . x))

        proof

          assume

           A19: ( omega -exponent (C6 . x)) in ( rng E4);

          then ( omega -exponent (C6 . x)) in ( dom (E4 " )) by FUNCT_1: 33;

          then x1 in ( rng (E4 " )) by FUNCT_1: 3;

          then x1 in ( dom E4) by FUNCT_1: 33;

          hence x1 in ( dom C4) by A1, Def1;

          

          hence ( omega -exponent (C4 . x1)) = (E4 . x1) by A1, Def1

          .= ( omega -exponent (C6 . x)) by A19, FUNCT_1: 35;

        end;

        

         A20: ( omega -exponent (C7 . x)) in ( rng E5) implies y2 in ( dom C5) & ( omega -exponent (C5 . y2)) = ( omega -exponent (C7 . x))

        proof

          assume

           A21: ( omega -exponent (C7 . x)) in ( rng E5);

          then ( omega -exponent (C7 . x)) in ( dom (E5 " )) by FUNCT_1: 33;

          then y2 in ( rng (E5 " )) by FUNCT_1: 3;

          then y2 in ( dom E5) by FUNCT_1: 33;

          hence y2 in ( dom C5) by A3, Def1;

          

          hence ( omega -exponent (C5 . y2)) = (E5 . y2) by A3, Def1

          .= ( omega -exponent (C7 . x)) by A21, FUNCT_1: 35;

        end;

        e in ( rng ( omega -exponent C6)) by A14, A16, FUNCT_1: 3;

        then e in (( rng E1) \/ ( rng E2)) or e in ( rng E3) by A1, A5, XBOOLE_0:def 3;

        per cases by XBOOLE_0:def 3;

          suppose

           A22: e in ( rng E1) & e in ( rng E2) & e in ( rng E3);

          then

           A23: e in (( rng E1) /\ ( rng E2)) & e in (( rng E2) /\ ( rng E3)) by XBOOLE_0:def 4;

          

           A24: e in ( rng E4) & e in ( rng E5) by A15, A22;

          then

           A25: e in (( rng E4) /\ ( rng E3)) & e in (( rng E1) /\ ( rng E5)) by A22, XBOOLE_0:def 4;

          

           A26: x1 in ( dom C4) & ( omega -exponent (C4 . x1)) = ( omega -exponent (C6 . x)) by A18, A24;

          

          then

           A27: (L4 . x1) = ( omega -leading_coeff (C4 . x1)) by A1, Def3

          .= ((L1 . ((E1 " ) . e)) + (L2 . ((E2 " ) . e))) by A2, A23, A26;

          

           A28: y2 in ( dom C5) & ( omega -exponent (C5 . y2)) = ( omega -exponent (C7 . x)) by A17, A20, A24;

          

          then

           A29: (L5 . y2) = ( omega -leading_coeff (C5 . y2)) by A3, Def3

          .= ((L2 . ((E2 " ) . ( omega -exponent (C7 . x)))) + (L3 . ((E3 " ) . ( omega -exponent (C7 . x))))) by A4, A17, A23, A28;

          ( omega -leading_coeff (C6 . x)) = ((L4 . x1) + (L3 . y1)) by A6, A13, A25

          .= ((L1 . ((E1 " ) . e)) + (L5 . y2)) by A17, A27, A29

          .= ( omega -leading_coeff (C7 . x)) by A8, A14, A17, A25;

          

          hence (C6 . x) = (( omega -leading_coeff (C7 . x)) *^ ( exp ( omega ,e))) by A13, Th64

          .= (C7 . x) by A14, A17, Th64;

        end;

          suppose

           A31: e in ( rng E1) & e in ( rng E2) & not e in ( rng E3);

          then

           A32: e in (( rng E1) /\ ( rng E2)) & e in (( rng E2) \ ( rng E3)) by XBOOLE_0:def 4, XBOOLE_0:def 5;

          

           A33: e in ( rng E4) & e in ( rng E5) by A15, A31;

          then

           A34: e in (( rng E4) \ ( rng E3)) & e in (( rng E1) /\ ( rng E5)) by A31, XBOOLE_0:def 4, XBOOLE_0:def 5;

          

           A35: x1 in ( dom C4) & ( omega -exponent (C4 . x1)) = ( omega -exponent (C6 . x)) by A18, A33;

          

          then

           A36: (L4 . x1) = ( omega -leading_coeff (C4 . x1)) by A1, Def3

          .= ((L1 . ((E1 " ) . e)) + (L2 . ((E2 " ) . e))) by A2, A32, A35;

          

           A37: y2 in ( dom C5) & ( omega -exponent (C5 . y2)) = ( omega -exponent (C7 . x)) by A17, A20, A33;

          

          then

           A38: (L5 . y2) = ( omega -leading_coeff (C5 . y2)) by A3, Def3

          .= (L2 . ((E2 " ) . ( omega -exponent (C7 . x)))) by A4, A17, A32, A37;

          ( omega -leading_coeff (C6 . x)) = (L4 . x1) by A6, A13, A34

          .= ( omega -leading_coeff (C7 . x)) by A8, A14, A17, A34, A36, A38;

          

          hence (C6 . x) = (( omega -leading_coeff (C7 . x)) *^ ( exp ( omega ,e))) by A13, Th64

          .= (C7 . x) by A14, A17, Th64;

        end;

          suppose

           A40: e in ( rng E1) & not e in ( rng E2) & e in ( rng E3);

          then

           A41: e in (( rng E1) \ ( rng E2)) & e in (( rng E3) \ ( rng E2)) by XBOOLE_0:def 5;

          

           A42: e in ( rng E4) & e in ( rng E5) by A15, A40;

          then

           A43: e in (( rng E4) /\ ( rng E3)) & e in (( rng E1) /\ ( rng E5)) by A40, XBOOLE_0:def 4;

          

           A44: x1 in ( dom C4) & ( omega -exponent (C4 . x1)) = ( omega -exponent (C6 . x)) by A18, A42;

          

          then

           A45: (L4 . x1) = ( omega -leading_coeff (C4 . x1)) by A1, Def3

          .= (L1 . ((E1 " ) . e)) by A2, A41, A44;

          

           A46: y2 in ( dom C5) & ( omega -exponent (C5 . y2)) = ( omega -exponent (C7 . x)) by A17, A20, A42;

          

          then

           A47: (L5 . y2) = ( omega -leading_coeff (C5 . y2)) by A3, Def3

          .= (L3 . ((E3 " ) . ( omega -exponent (C7 . x)))) by A4, A17, A41, A46;

          ( omega -leading_coeff (C6 . x)) = ((L4 . x1) + (L3 . y1)) by A6, A13, A43

          .= ( omega -leading_coeff (C7 . x)) by A8, A14, A17, A43, A45, A47;

          

          hence (C6 . x) = (( omega -leading_coeff (C7 . x)) *^ ( exp ( omega ,e))) by A13, Th64

          .= (C7 . x) by A14, A17, Th64;

        end;

          suppose

           A49: not e in ( rng E1) & e in ( rng E2) & e in ( rng E3);

          then

           A50: e in (( rng E2) \ ( rng E1)) & e in (( rng E2) /\ ( rng E3)) by XBOOLE_0:def 4, XBOOLE_0:def 5;

          

           A51: e in ( rng E4) & e in ( rng E5) by A15, A49;

          then

           A52: e in (( rng E4) /\ ( rng E3)) & e in (( rng E5) \ ( rng E1)) by A49, XBOOLE_0:def 4, XBOOLE_0:def 5;

          

           A53: x1 in ( dom C4) & ( omega -exponent (C4 . x1)) = ( omega -exponent (C6 . x)) by A18, A51;

          

          then

           A54: (L4 . x1) = ( omega -leading_coeff (C4 . x1)) by A1, Def3

          .= (L2 . ((E2 " ) . e)) by A2, A50, A53;

          

           A55: y2 in ( dom C5) & ( omega -exponent (C5 . y2)) = ( omega -exponent (C7 . x)) by A17, A20, A51;

          

          then

           A56: (L5 . y2) = ( omega -leading_coeff (C5 . y2)) by A3, Def3

          .= ((L2 . ((E2 " ) . ( omega -exponent (C7 . x)))) + (L3 . ((E3 " ) . ( omega -exponent (C7 . x))))) by A4, A17, A50, A55;

          

           A57: ( omega -leading_coeff (C6 . x)) = ((L2 . ((E2 " ) . e)) + (L3 . y1)) by A6, A13, A52, A54

          .= ( omega -leading_coeff (C7 . x)) by A8, A14, A17, A52, A56;

          

          thus (C6 . x) = (( omega -leading_coeff (C6 . x)) *^ ( exp ( omega ,e))) by A13, Th64

          .= (C7 . x) by A14, A17, A57, Th64;

        end;

          suppose

           A58: e in ( rng E1) & not e in ( rng E2) & not e in ( rng E3);

          then

           A59: e in (( rng E1) \ ( rng E2)) & not e in (( rng E2) \/ ( rng E3)) by XBOOLE_0:def 3, XBOOLE_0:def 5;

          then

           A60: e in ( rng E4) & not e in ( rng E5) by A10, A15, TARSKI:def 3;

          then

           A61: e in (( rng E4) \ ( rng E3)) & e in (( rng E1) \ ( rng E5)) by A58, XBOOLE_0:def 5;

          

           A62: x1 in ( dom C4) & ( omega -exponent (C4 . x1)) = ( omega -exponent (C6 . x)) by A18, A60;

          

          then

           A63: (L4 . x1) = ( omega -leading_coeff (C4 . x1)) by A1, Def3

          .= (L1 . ((E1 " ) . e)) by A2, A59, A62;

          ( omega -leading_coeff (C6 . x)) = (L4 . x1) by A6, A13, A61

          .= ( omega -leading_coeff (C7 . x)) by A8, A14, A17, A61, A63;

          

          hence (C6 . x) = (( omega -leading_coeff (C7 . x)) *^ ( exp ( omega ,e))) by A13, Th64

          .= (C7 . x) by A14, A17, Th64;

        end;

          suppose

           A65: not e in ( rng E1) & e in ( rng E2) & not e in ( rng E3);

          then

           A66: e in (( rng E2) \ ( rng E1)) & e in (( rng E2) \ ( rng E3)) by XBOOLE_0:def 5;

          

           A67: e in ( rng E4) & e in ( rng E5) by A15, A65;

          then

           A68: e in (( rng E4) \ ( rng E3)) & e in (( rng E5) \ ( rng E1)) by A65, XBOOLE_0:def 5;

          

           A69: x1 in ( dom C4) & ( omega -exponent (C4 . x1)) = ( omega -exponent (C6 . x)) by A18, A67;

          

          then

           A70: (L4 . x1) = ( omega -leading_coeff (C4 . x1)) by A1, Def3

          .= (L2 . ((E2 " ) . e)) by A2, A66, A69;

          

           A71: y2 in ( dom C5) & ( omega -exponent (C5 . y2)) = ( omega -exponent (C7 . x)) by A17, A20, A67;

          

          then

           A72: (L5 . y2) = ( omega -leading_coeff (C5 . y2)) by A3, Def3

          .= (L2 . ((E2 " ) . ( omega -exponent (C7 . x)))) by A4, A17, A66, A71;

          ( omega -leading_coeff (C6 . x)) = (L2 . ((E2 " ) . e)) by A6, A13, A68, A70

          .= ( omega -leading_coeff (C7 . x)) by A8, A14, A17, A68, A72;

          

          hence (C6 . x) = (( omega -leading_coeff (C7 . x)) *^ ( exp ( omega ,e))) by A13, Th64

          .= (C7 . x) by A14, A17, Th64;

        end;

          suppose

           A74: not e in ( rng E1) & not e in ( rng E2) & e in ( rng E3);

          then

           A75: not e in (( rng E1) \/ ( rng E2)) & e in (( rng E3) \ ( rng E2)) by XBOOLE_0:def 3, XBOOLE_0:def 5;

          then

           A76: not e in ( rng E4) & e in ( rng E5) by A9, A15, TARSKI:def 3;

          then

           A77: e in (( rng E3) \ ( rng E4)) & e in (( rng E5) \ ( rng E1)) by A74, XBOOLE_0:def 5;

          

           A78: y2 in ( dom C5) & ( omega -exponent (C5 . y2)) = ( omega -exponent (C7 . x)) by A17, A20, A76;

          

          then

           A79: (L5 . y2) = ( omega -leading_coeff (C5 . y2)) by A3, Def3

          .= (L3 . ((E3 " ) . ( omega -exponent (C7 . x)))) by A4, A17, A75, A78;

          ( omega -leading_coeff (C6 . x)) = (L3 . y1) by A6, A13, A77

          .= ( omega -leading_coeff (C7 . x)) by A8, A14, A17, A77, A79;

          

          hence (C6 . x) = (( omega -leading_coeff (C7 . x)) *^ ( exp ( omega ,e))) by A13, Th64

          .= (C7 . x) by A14, A17, Th64;

        end;

      end;

      hence thesis by A5, A7, A12, FUNCT_1: 2;

    end;

    theorem :: ORDINAL7:69

    

     Th82: for a be Ordinal holds (a (+) 0 ) = a

    proof

      let a be Ordinal;

      set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF 0 ));

      set L1 = ( omega -leading_coeff ( CantorNF a));

      set L2 = ( omega -leading_coeff ( CantorNF 0 ));

      consider C be Cantor-normal-form Ordinal-Sequence such that

       A1: (a (+) 0 ) = ( Sum^ C) & ( rng ( omega -exponent C)) = (( rng E1) \/ ( rng E2)) and

       A2: for d be object st d in ( dom C) holds (( omega -exponent (C . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C . d)) = (L1 . ((E1 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C . d)) = (L2 . ((E2 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d)))))) by Def5;

      

       A3: ( rng E2) is empty;

      then

       A4: ( rng ( omega -exponent C)) = ( rng E1) by A1;

      

       A5: ( dom C) = ( card ( dom ( omega -exponent C))) by Def1

      .= ( card ( rng ( omega -exponent C))) by CARD_1: 70

      .= ( card ( dom E1)) by A4, CARD_1: 70

      .= ( dom ( CantorNF a)) by Def1;

      for x be object st x in ( dom C) holds (C . x) = (( CantorNF a) . x)

      proof

        let x be object;

        

         A6: ( omega -exponent C) = E1 by A4, Th34;

        assume

         A7: x in ( dom C);

        then

         A8: x in ( dom ( omega -exponent C)) by Def1;

        then (( omega -exponent C) . x) in ( rng E1) by A4, FUNCT_1: 3;

        then ( omega -exponent (C . x)) in (( rng E1) \ ( rng E2)) by A3, A7, Def1;

        

        then

         A9: ( omega -leading_coeff (C . x)) = (L1 . ((E1 " ) . ( omega -exponent (C . x)))) by A2, A7

        .= (L1 . ((E1 " ) . (( omega -exponent C) . x))) by A7, Def1

        .= (L1 . x) by A6, A8, FUNCT_1: 34;

        

         A10: x in ( dom ( CantorNF a)) by A6, A8, Def1;

        

        thus (C . x) = ((L1 . x) *^ ( exp ( omega ,( omega -exponent (C . x))))) by A7, A9, Th64

        .= ((L1 . x) *^ ( exp ( omega ,(E1 . x)))) by A6, A7, Def1

        .= ((L1 . x) *^ ( exp ( omega ,( omega -exponent (( CantorNF a) . x))))) by A10, Def1

        .= (( omega -leading_coeff (( CantorNF a) . x)) *^ ( exp ( omega ,( omega -exponent (( CantorNF a) . x))))) by A10, Def3

        .= (( CantorNF a) . x) by A10, Th64;

      end;

      then C = ( CantorNF a) by A5, FUNCT_1: 2;

      hence thesis by A1;

    end;

    theorem :: ORDINAL7:70

    

     Th83: for a,b be Ordinal, n be Nat st ( omega -exponent a) c= b holds ((n *^ ( exp ( omega ,b))) (+) a) = ((n *^ ( exp ( omega ,b))) +^ a)

    proof

      let a,b be Ordinal, n be Nat;

      set E1 = ( omega -exponent ( CantorNF (n *^ ( exp ( omega ,b)))));

      set E2 = ( omega -exponent ( CantorNF a));

      set L1 = ( omega -leading_coeff ( CantorNF (n *^ ( exp ( omega ,b)))));

      set L2 = ( omega -leading_coeff ( CantorNF a));

      assume ( omega -exponent a) c= b;

      then ( omega -exponent ( Sum^ ( CantorNF a))) c= b;

      then

       A1: ( omega -exponent (( CantorNF a) . 0 )) c= b by Th44;

      

       A2: (E2 . 0 ) c= b

      proof

        per cases ;

          suppose 0 in ( dom ( CantorNF a));

          hence thesis by A1, Def1;

        end;

          suppose not 0 in ( dom ( CantorNF a));

          then not 0 in ( dom E2) by Def1;

          then (E2 . 0 ) = {} by FUNCT_1:def 2;

          hence thesis;

        end;

      end;

      consider C be Cantor-normal-form Ordinal-Sequence such that

       A3: ((n *^ ( exp ( omega ,b))) (+) a) = ( Sum^ C) and

       A4: ( rng ( omega -exponent C)) = (( rng E1) \/ ( rng E2)) and

       A5: for d be object st d in ( dom C) holds (( omega -exponent (C . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C . d)) = (L1 . ((E1 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C . d)) = (L2 . ((E2 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d)))))) by Def5;

      per cases ;

        suppose

         A6: a = 0 ;

        

        hence ((n *^ ( exp ( omega ,b))) (+) a) = (n *^ ( exp ( omega ,b))) by Th82

        .= ((n *^ ( exp ( omega ,b))) +^ a) by A6, ORDINAL2: 27;

      end;

        suppose

         A7: n is zero;

        

        hence ((n *^ ( exp ( omega ,b))) (+) a) = ( 0 (+) a) by ORDINAL2: 35

        .= a by Th82

        .= ( 0 +^ a) by ORDINAL2: 30

        .= ((n *^ ( exp ( omega ,b))) +^ a) by A7, ORDINAL2: 35;

      end;

        suppose

         A8: n is non zero & a <> 0 & (E2 . 0 ) = b;

        then

        consider a0 be Cantor-component Ordinal, A0 be Cantor-normal-form Ordinal-Sequence such that

         A9: ( CantorNF a) = ( <%a0%> ^ A0) by ORDINAL5: 67;

         0 c< n by A8, XBOOLE_1: 2, XBOOLE_0:def 8;

        then

         A10: 0 in n & n in omega by ORDINAL1: 11, ORDINAL1:def 12;

        

         A11: E1 = ( omega -exponent <%(n *^ ( exp ( omega ,b)))%>) by A8, Th69

        .= <%( omega -exponent (n *^ ( exp ( omega ,b))))%> by Th46

        .= <%b%> by A10, ORDINAL5: 58;

        then

         A12: ( rng E1) = {b} by AFINSQ_1: 33;

         0 c< ( dom ( CantorNF a)) by A8, XBOOLE_1: 2, XBOOLE_0:def 8;

        then

         A13: 0 in ( dom ( CantorNF a)) by ORDINAL1: 11;

        then

         A14: 0 in ( dom E2) by Def1;

        then ( rng E1) c= ( rng E2) by A8, A12, FUNCT_1: 3, ZFMISC_1: 31;

        then

         A15: ( omega -exponent C) = E2 by A4, Th34, XBOOLE_1: 12;

        

         A16: ( dom C) = ( card ( dom ( omega -exponent C))) by Def1

        .= ( len ( <%a0%> ^ A0)) by A15, A9, Def1

        .= (( len <%a0%>) + ( len A0)) by AFINSQ_1: 17

        .= (1 + ( len A0)) by AFINSQ_1: 34

        .= (( len <%((n *^ ( exp ( omega ,b))) +^ a0)%>) + ( len A0)) by AFINSQ_1: 34

        .= ( dom ( <%((n *^ ( exp ( omega ,b))) +^ a0)%> ^ A0)) by AFINSQ_1: 17;

        for x be object st x in ( dom C) holds (C . x) = (( <%((n *^ ( exp ( omega ,b))) +^ a0)%> ^ A0) . x)

        proof

          let x be object;

          assume

           A17: x in ( dom C);

          

           A18: ( dom C) = ( dom E2) by Def1, A15;

          per cases ;

            suppose

             A19: ( omega -exponent (C . x)) = b;

            then b = (E2 . x) by A15, A17, Def1;

            then

             A20: x = 0 by A8, A14, A17, A18, FUNCT_1:def 4;

            

             A21: ( omega -exponent (C . x)) in ( rng E2) by A8, A14, A19, FUNCT_1: 3;

            ( omega -exponent (C . x)) in ( rng E1) by A12, A19, TARSKI:def 1;

            then

             A22: ( omega -exponent (C . x)) in (( rng E1) /\ ( rng E2)) by A21, XBOOLE_0:def 4;

            

             A23: (E1 . 0 ) = b by A11;

            ( dom E1) = 1 by A11, AFINSQ_1: 34;

            then

             A24: 0 in ( dom E1) by CARD_1: 49, TARSKI:def 1;

            

             A25: ( omega -leading_coeff (C . x)) = ((L1 . ((E1 " ) . ( omega -exponent (C . x)))) + (L2 . ((E2 " ) . b))) by A5, A17, A19, A22

            .= ((L1 . 0 ) + (L2 . ((E2 " ) . b))) by A19, A23, A24, FUNCT_1: 34

            .= ((L1 . 0 ) + (L2 . 0 )) by A8, A14, FUNCT_1: 34

            .= ((L1 . 0 ) +^ (L2 . 0 )) by CARD_2: 36;

            

             A26: (L1 . 0 ) = (( omega -leading_coeff <%(n *^ ( exp ( omega ,b)))%>) . 0 ) by A8, Th69

            .= ( <%( omega -leading_coeff (n *^ ( exp ( omega ,b))))%> . 0 ) by Th60

            .= n by Th57, ORDINAL1:def 12;

            

            thus (C . x) = (((L1 . 0 ) +^ (L2 . 0 )) *^ ( exp ( omega ,( omega -exponent (C . x))))) by A17, A25, Th64

            .= (((L1 . 0 ) *^ ( exp ( omega ,b))) +^ ((L2 . 0 ) *^ ( exp ( omega ,(E2 . 0 ))))) by A8, A19, ORDINAL3: 46

            .= ((n *^ ( exp ( omega ,b))) +^ (( CantorNF a) . 0 )) by A13, A26, Th65

            .= ((n *^ ( exp ( omega ,b))) +^ a0) by A9, AFINSQ_1: 35

            .= (( <%((n *^ ( exp ( omega ,b))) +^ a0)%> ^ A0) . x) by A20, AFINSQ_1: 35;

          end;

            suppose

             A27: ( omega -exponent (C . x)) <> b;

            then

             A28: not ( omega -exponent (C . x)) in ( rng E1) by A12, TARSKI:def 1;

            x in ( dom ( omega -exponent C)) by A17, Def1;

            then (( omega -exponent C) . x) in ( rng ( omega -exponent C)) by FUNCT_1: 3;

            then ( omega -exponent (C . x)) in (( rng E1) \/ ( rng E2)) by A4, A17, Def1;

            then ( omega -exponent (C . x)) in ( rng E1) or ( omega -exponent (C . x)) in ( rng E2) by XBOOLE_0:def 3;

            then ( omega -exponent (C . x)) in (( rng E2) \ ( rng E1)) by A28, XBOOLE_0:def 5;

            

            then

             A29: ( omega -leading_coeff (C . x)) = (L2 . ((E2 " ) . ( omega -exponent (C . x)))) by A5, A17

            .= (L2 . ((E2 " ) . (( omega -exponent C) . x))) by A17, Def1

            .= (L2 . x) by A15, A17, A18, FUNCT_1: 34;

            (( omega -exponent C) . x) <> b by A17, A27, Def1;

            then not x in 1 by A8, A15, CARD_1: 49, TARSKI:def 1;

            then not x in ( len <%((n *^ ( exp ( omega ,b))) +^ a0)%>) by AFINSQ_1: 34;

            then

            consider m be Nat such that

             A30: m in ( dom A0) & x = (( len <%((n *^ ( exp ( omega ,b))) +^ a0)%>) + m) by A16, A17, AFINSQ_1: 20;

            

             A31: x = (1 + m) by A30, AFINSQ_1: 34

            .= (( len <%a0%>) + m) by AFINSQ_1: 34;

            

             A32: x in ( dom ( CantorNF a)) by A17, A18, Def1;

            

            thus (C . x) = ((L2 . x) *^ ( exp ( omega ,( omega -exponent (C . x))))) by A17, A29, Th64

            .= ((L2 . x) *^ ( exp ( omega ,(( omega -exponent C) . x)))) by A17, Def1

            .= (( CantorNF a) . x) by A15, A32, Th65

            .= (A0 . m) by A9, A30, A31, AFINSQ_1:def 3

            .= (( <%((n *^ ( exp ( omega ,b))) +^ a0)%> ^ A0) . x) by A30, AFINSQ_1:def 3;

          end;

        end;

        then C = ( <%((n *^ ( exp ( omega ,b))) +^ a0)%> ^ A0) by A16, FUNCT_1: 2;

        

        hence ((n *^ ( exp ( omega ,b))) (+) a) = (((n *^ ( exp ( omega ,b))) +^ a0) +^ ( Sum^ A0)) by A3, ORDINAL5: 55

        .= ((n *^ ( exp ( omega ,b))) +^ (a0 +^ ( Sum^ A0))) by ORDINAL3: 30

        .= ((n *^ ( exp ( omega ,b))) +^ ( Sum^ ( <%a0%> ^ A0))) by ORDINAL5: 55

        .= ((n *^ ( exp ( omega ,b))) +^ a) by A9;

      end;

        suppose

         A33: n is non zero & a <> 0 & (E2 . 0 ) <> b;

        then

         A34: (E2 . 0 ) in b by A2, XBOOLE_0:def 8, ORDINAL1: 11;

         0 c< n by A33, XBOOLE_1: 2, XBOOLE_0:def 8;

        then

         A35: 0 in n & n in omega by ORDINAL1: 11, ORDINAL1:def 12;

        

         A36: E1 = ( omega -exponent <%(n *^ ( exp ( omega ,b)))%>) by A33, Th69

        .= <%( omega -exponent (n *^ ( exp ( omega ,b))))%> by Th46

        .= <%b%> by A35, ORDINAL5: 58;

        then

         A37: ( rng E1) = {b} by AFINSQ_1: 33;

        (( rng E1) /\ ( rng E2)) = {}

        proof

          assume (( rng E1) /\ ( rng E2)) <> {} ;

          then

          consider y be object such that

           A38: y in (( rng E1) /\ ( rng E2)) by XBOOLE_0:def 1;

          

           A39: y in ( rng E1) & y in ( rng E2) by A38, XBOOLE_0:def 4;

          then y = b by A37, TARSKI:def 1;

          then

          consider x be object such that

           A40: x in ( dom E2) & (E2 . x) = b by A39, FUNCT_1:def 3;

          reconsider x as Ordinal by A40;

          x <> 0 by A34, A40;

          then 0 c< x by XBOOLE_1: 2, XBOOLE_0:def 8;

          then

           A41: 0 in x by ORDINAL1: 11;

          

           A42: x in ( dom ( CantorNF a)) by A40, Def1;

          then ( omega -exponent (( CantorNF a) . x)) in ( omega -exponent (( CantorNF a) . 0 )) by A41, ORDINAL5:def 11;

          then b in ( omega -exponent (( CantorNF a) . 0 )) by A40, A42, Def1;

          hence contradiction by A34, A41, A42, Def1, ORDINAL1: 10;

        end;

        then

         A43: ( rng E1) misses ( rng E2) by XBOOLE_0:def 7;

        

         A44: ( dom C) = ( card ( dom ( omega -exponent C))) by Def1

        .= ( card ( rng ( omega -exponent C))) by CARD_1: 70

        .= (( card ( rng E1)) +` ( card ( rng E2))) by A4, A43, CARD_2: 35

        .= (( card ( dom E1)) +` ( card ( rng E2))) by CARD_1: 70

        .= (( dom E1) +` ( card ( dom E2))) by CARD_1: 70

        .= (( len E1) + ( dom ( CantorNF a))) by Def1

        .= (1 + ( dom ( CantorNF a))) by A36, AFINSQ_1: 34

        .= (( len <%(n *^ ( exp ( omega ,b)))%>) + ( len ( CantorNF a))) by AFINSQ_1: 34

        .= ( dom ( <%(n *^ ( exp ( omega ,b)))%> ^ ( CantorNF a))) by AFINSQ_1: 17;

        for x be object st x in ( dom C) holds (C . x) = (( <%(n *^ ( exp ( omega ,b)))%> ^ ( CantorNF a)) . x)

        proof

          let x be object;

          assume

           A45: x in ( dom C);

          for c,d be Ordinal st c in d & d in ( dom (E1 ^ E2)) holds ((E1 ^ E2) . d) in ((E1 ^ E2) . c)

          proof

            let c,d be Ordinal;

            assume

             A46: c in d & d in ( dom (E1 ^ E2));

            then

             A47: c in ( dom (E1 ^ E2)) by ORDINAL1: 10;

            then

            reconsider m1 = c, m2 = d as Nat by A46;

            per cases by A47, AFINSQ_1: 20;

              suppose

               A48: m1 in ( dom E1);

              then m1 in 1 by A36, AFINSQ_1: 34;

              then

               A49: m1 = 0 by CARD_1: 49, TARSKI:def 1;

              

               A50: ((E1 ^ E2) . m1) = (E1 . m1) by A48, AFINSQ_1:def 3

              .= b by A36, A49;

               not m2 in ( dom E1)

              proof

                assume m2 in ( dom E1);

                then m2 in 1 by A36, AFINSQ_1: 34;

                hence contradiction by A46, CARD_1: 49, TARSKI:def 1;

              end;

              then

              consider k2 be Nat such that

               A51: k2 in ( dom E2) & m2 = (( len E1) + k2) by A46, AFINSQ_1: 20;

              

               A52: ((E1 ^ E2) . m2) = (E2 . k2) by A51, AFINSQ_1:def 3;

              per cases ;

                suppose k2 = 0 ;

                hence thesis by A34, A50, A52;

              end;

                suppose k2 <> 0 ;

                then 0 c< k2 by XBOOLE_1: 2, XBOOLE_0:def 8;

                then 0 in k2 by ORDINAL1: 11;

                then (E2 . k2) in (E2 . 0 ) by A51, ORDINAL5:def 1;

                hence thesis by A34, A50, A52, ORDINAL1: 10;

              end;

            end;

              suppose ex k1 be Nat st k1 in ( dom E2) & m1 = (( len E1) + k1);

              then

              consider k1 be Nat such that

               A53: k1 in ( dom E2) & m1 = (( len E1) + k1);

              

               A54: ((E1 ^ E2) . m1) = (E2 . k1) by A53, AFINSQ_1:def 3;

               not m2 in ( dom E1)

              proof

                assume m2 in ( dom E1);

                then

                 A55: m2 in 1 by A36, AFINSQ_1: 34;

                per cases ;

                  suppose k1 = 0 ;

                  hence contradiction by A46, A36, A53, A55, AFINSQ_1: 34;

                end;

                  suppose k1 <> 0 ;

                  then ( len E1) < (( len E1) + k1) by NAT_1: 16;

                  then 1 < m1 by A36, A53, AFINSQ_1: 34;

                  then 1 in ( Segm m1) by NAT_1: 44;

                  hence contradiction by A46, A55, ORDINAL1: 10;

                end;

              end;

              then

              consider k2 be Nat such that

               A56: k2 in ( dom E2) & m2 = (( len E1) + k2) by A46, AFINSQ_1: 20;

              

               A57: ((E1 ^ E2) . m2) = (E2 . k2) by A56, AFINSQ_1:def 3;

              m1 in ( Segm m2) by A46;

              then ((( len E1) + k1) - ( len E1)) < ((( len E1) + k2) - ( len E1)) by A53, A56, NAT_1: 44, XREAL_1: 14;

              then k1 in ( Segm k2) by NAT_1: 44;

              hence thesis by A54, A56, A57, ORDINAL5:def 1;

            end;

          end;

          then

           A58: (E1 ^ E2) is decreasing by ORDINAL5:def 1;

          ( rng (E1 ^ E2)) = ( rng ( omega -exponent C)) by A4, AFINSQ_1: 26;

          then

           A59: ( omega -exponent C) = (E1 ^ E2) by A58, Th34;

          per cases ;

            suppose

             A60: x = 0 ;

            

             A61: ( omega -exponent (C . x)) = (( omega -exponent C) . x) by A45, Def1

            .= b by A36, A59, A60, AFINSQ_1: 35;

            then ( omega -exponent (C . x)) in ( rng E1) by A37, TARSKI:def 1;

            then

             A62: ( omega -exponent (C . x)) in (( rng E1) \ ( rng E2)) by A43, XBOOLE_1: 83;

            

             A63: (E1 . 0 ) = b by A36;

            ( dom E1) = 1 by A36, AFINSQ_1: 34;

            then

             A64: 0 in ( dom E1) by CARD_1: 49, TARSKI:def 1;

            ( omega -leading_coeff (C . x)) = (L1 . ((E1 " ) . ( omega -exponent (C . x)))) by A5, A45, A62

            .= (L1 . 0 ) by A61, A63, A64, FUNCT_1: 34

            .= (( omega -leading_coeff <%(n *^ ( exp ( omega ,b)))%>) . 0 ) by A33, Th69

            .= ( <%( omega -leading_coeff (n *^ ( exp ( omega ,b))))%> . 0 ) by Th60

            .= n by Th57, ORDINAL1:def 12;

            

            hence (C . x) = (n *^ ( exp ( omega ,b))) by A45, A61, Th64

            .= (( <%(n *^ ( exp ( omega ,b)))%> ^ ( CantorNF a)) . x) by A60, AFINSQ_1: 35;

          end;

            suppose

             A66: x <> 0 ;

            then not x in 1 by CARD_1: 49, TARSKI:def 1;

            then

             A67: not x in ( len E1) by A36, AFINSQ_1: 34;

            

             A68: x in ( dom ( omega -exponent C)) by A45, Def1;

            then

            consider k be Nat such that

             A69: k in ( dom E2) & x = (( len E1) + k) by A59, A67, AFINSQ_1: 20;

            ( omega -exponent (C . x)) <> b

            proof

               0 in 1 by CARD_1: 49, TARSKI:def 1;

              then

               A70: 0 in ( dom E1) by A36, AFINSQ_1: 34;

              assume ( omega -exponent (C . x)) = b;

              

              then

               A71: (( omega -exponent C) . x) = (E1 . 0 ) by A36, A45, Def1

              .= (( omega -exponent C) . 0 ) by A59, A70, AFINSQ_1:def 3;

               0 in ( dom ( omega -exponent C)) by A59, A70, TARSKI:def 3, AFINSQ_1: 21;

              hence contradiction by A66, A68, A71, FUNCT_1:def 4;

            end;

            then

             A72: not ( omega -exponent (C . x)) in ( rng E1) by A37, TARSKI:def 1;

            

             A73: k in ( dom ( CantorNF a)) by A69, Def1;

            

             A74: x = (1 + k) by A36, A69, AFINSQ_1: 34

            .= (( len <%(n *^ ( exp ( omega ,b)))%>) + k) by AFINSQ_1: 34;

            x in ( dom ( omega -exponent C)) by A45, Def1;

            then (( omega -exponent C) . x) in ( rng ( omega -exponent C)) by FUNCT_1: 3;

            then ( omega -exponent (C . x)) in (( rng E1) \/ ( rng E2)) by A4, A45, Def1;

            then ( omega -exponent (C . x)) in ( rng E2) by A72, XBOOLE_0:def 3;

            then ( omega -exponent (C . x)) in (( rng E2) \ ( rng E1)) by A72, XBOOLE_0:def 5;

            

            then ( omega -leading_coeff (C . x)) = (L2 . ((E2 " ) . ( omega -exponent (C . x)))) by A5, A45

            .= (L2 . ((E2 " ) . (( omega -exponent C) . x))) by A45, Def1

            .= (L2 . ((E2 " ) . (E2 . k))) by A59, A69, AFINSQ_1:def 3

            .= (L2 . k) by A69, FUNCT_1: 34;

            

            hence (C . x) = ((L2 . k) *^ ( exp ( omega ,( omega -exponent (C . x))))) by A45, Th64

            .= ((L2 . k) *^ ( exp ( omega ,(( omega -exponent C) . x)))) by A45, Def1

            .= ((L2 . k) *^ ( exp ( omega ,(E2 . k)))) by A59, A69, AFINSQ_1:def 3

            .= (( CantorNF a) . k) by A73, Th65

            .= (( <%(n *^ ( exp ( omega ,b)))%> ^ ( CantorNF a)) . x) by A73, A74, AFINSQ_1:def 3;

          end;

        end;

        

        hence ((n *^ ( exp ( omega ,b))) (+) a) = ( Sum^ ( <%(n *^ ( exp ( omega ,b)))%> ^ ( CantorNF a))) by A3, A44, FUNCT_1: 2

        .= ((n *^ ( exp ( omega ,b))) +^ ( Sum^ ( CantorNF a))) by ORDINAL5: 55

        .= ((n *^ ( exp ( omega ,b))) +^ a);

      end;

    end;

    theorem :: ORDINAL7:71

    

     Th84: for A,B be finite Ordinal-Sequence st (A ^ B) is Cantor-normal-form holds (( Sum^ A) (+) ( Sum^ B)) = (( Sum^ A) +^ ( Sum^ B))

    proof

      defpred P[ Nat] means for A,B be finite Ordinal-Sequence st ( len A) = $1 & (A ^ B) is Cantor-normal-form holds (( Sum^ A) (+) ( Sum^ B)) = (( Sum^ A) +^ ( Sum^ B));

      

       A1: P[ 0 ]

      proof

        let A,B be finite Ordinal-Sequence;

        assume ( len A) = 0 & (A ^ B) is Cantor-normal-form;

        then A is empty;

        then

         A2: ( Sum^ A) = 0 by ORDINAL5: 52;

        

        hence (( Sum^ A) (+) ( Sum^ B)) = ( Sum^ B) by Th82

        .= (( Sum^ A) +^ ( Sum^ B)) by A2, ORDINAL2: 30;

      end;

      

       A3: for n be Nat st P[n] holds P[(n + 1)]

      proof

        let n be Nat;

        assume

         A4: P[n];

        let A,B be finite Ordinal-Sequence;

        assume

         A5: ( len A) = (n + 1) & (A ^ B) is Cantor-normal-form;

        then

         A6: A <> {} & A is Cantor-normal-form by ORDINAL5: 66;

        then

        consider a0 be Cantor-component Ordinal, A0 be Cantor-normal-form Ordinal-Sequence such that

         A7: A = ( <%a0%> ^ A0) by ORDINAL5: 67;

        

         A8: ( <%a0%> ^ (A0 ^ B)) is Cantor-normal-form by A5, A7, AFINSQ_1: 27;

        then

         A9: (A0 ^ B) is Cantor-normal-form by ORDINAL5: 66;

        (n + 1) = (( len <%a0%>) + ( len A0)) by A5, A7, AFINSQ_1: 17

        .= (1 + ( len A0)) by AFINSQ_1: 34;

        then

         A10: (( Sum^ A0) (+) ( Sum^ B)) = (( Sum^ A0) +^ ( Sum^ B)) by A4, A9;

        consider b be Ordinal, m be Nat such that

         A11: 0 in ( Segm m) & a0 = (m *^ ( exp ( omega ,b))) by ORDINAL5:def 9;

        reconsider m as non zero Nat by A11;

         0 in m & m in omega by A11, ORDINAL1:def 12;

        then

         A12: ( omega -exponent a0) = b by A11, ORDINAL5: 58;

        

         A13: ( omega -exponent ( Sum^ A0)) c= b

        proof

          per cases ;

            suppose

             A14: 0 in ( Sum^ A0);

            ( Sum^ A0) in ( exp ( omega ,( omega -exponent a0))) by A6, A7, Th43;

            hence thesis by A12, A14, Th23, ORDINAL1:def 2;

          end;

            suppose not 0 in ( Sum^ A0);

            then ( omega -exponent ( Sum^ A0)) = 0 by ORDINAL5:def 10;

            hence thesis;

          end;

        end;

        

         A15: ( omega -exponent (( Sum^ A0) (+) ( Sum^ B))) c= b

        proof

          

           A16: (( Sum^ A0) (+) ( Sum^ B)) = ( Sum^ (A0 ^ B)) by A10, Th24;

          per cases ;

            suppose

             A17: 0 in ( Sum^ (A0 ^ B));

            ( Sum^ (A0 ^ B)) in ( exp ( omega ,( omega -exponent a0))) by A8, Th43;

            hence thesis by A12, A16, A17, Th23, ORDINAL1:def 2;

          end;

            suppose not 0 in ( Sum^ (A0 ^ B));

            then ( omega -exponent ( Sum^ (A0 ^ B))) = 0 by ORDINAL5:def 10;

            hence thesis by A16;

          end;

        end;

        

        thus (( Sum^ A) (+) ( Sum^ B)) = ((a0 +^ ( Sum^ A0)) (+) ( Sum^ B)) by A7, ORDINAL5: 55

        .= ((a0 (+) ( Sum^ A0)) (+) ( Sum^ B)) by A11, A13, Th83

        .= (a0 (+) (( Sum^ A0) (+) ( Sum^ B))) by Th81

        .= (a0 +^ (( Sum^ A0) (+) ( Sum^ B))) by A11, A15, Th83

        .= ((a0 +^ ( Sum^ A0)) +^ ( Sum^ B)) by A10, ORDINAL3: 30

        .= (( Sum^ A) +^ ( Sum^ B)) by A7, ORDINAL5: 55;

      end;

      

       A18: for n be Nat holds P[n] from NAT_1:sch 2( A1, A3);

      let A,B be finite Ordinal-Sequence;

      assume

       A19: (A ^ B) is Cantor-normal-form;

      ( len A) is Nat;

      hence thesis by A18, A19;

    end;

    theorem :: ORDINAL7:72

    

     Th85: for a,b be Ordinal st a <> 0 implies ( omega -exponent b) in ( omega -exponent ( last ( CantorNF a))) holds (a (+) b) = (a +^ b)

    proof

      let a,b be Ordinal;

      assume

       A1: a <> 0 implies ( omega -exponent b) in ( omega -exponent ( last ( CantorNF a)));

      per cases ;

        suppose a = 0 ;

        then (a (+) b) = b & (a +^ b) = b by Th82, ORDINAL2: 30;

        hence thesis;

      end;

        suppose b = 0 ;

        then (a (+) b) = a & (a +^ b) = a by Th82, ORDINAL2: 27;

        hence thesis;

      end;

        suppose

         A2: a <> 0 & b <> 0 ;

        then ( omega -exponent ( Sum^ ( CantorNF b))) in ( omega -exponent ( last ( CantorNF a))) by A1;

        then ( omega -exponent (( CantorNF b) . 0 )) in ( omega -exponent ( last ( CantorNF a))) by Th44;

        then

         A3: (( CantorNF a) ^ ( CantorNF b)) is Cantor-normal-form by A2, Th33;

        

        thus (a (+) b) = (( Sum^ ( CantorNF a)) (+) ( Sum^ ( CantorNF b)))

        .= (a +^ b) by A3, Th84;

      end;

    end;

    theorem :: ORDINAL7:73

    

     Th86: for a,b be Ordinal, n be Nat st a <> 0 implies b c= ( omega -exponent ( last ( CantorNF a))) holds (a (+) (n *^ ( exp ( omega ,b)))) = (a +^ (n *^ ( exp ( omega ,b))))

    proof

      let a,b be Ordinal, n be Nat;

      set c = (n *^ ( exp ( omega ,b)));

      set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF c));

      set L1 = ( omega -leading_coeff ( CantorNF a));

      set L2 = ( omega -leading_coeff ( CantorNF c));

      assume

       A1: a <> 0 implies b c= ( omega -exponent ( last ( CantorNF a)));

      per cases ;

        suppose

         A2: a = 0 ;

        

        hence (a (+) (n *^ ( exp ( omega ,b)))) = (n *^ ( exp ( omega ,b))) by Th82

        .= (a +^ (n *^ ( exp ( omega ,b)))) by A2, ORDINAL2: 30;

      end;

        suppose not 0 in n;

        then

         A3: n = 0 by ORDINAL1: 16, XBOOLE_1: 3;

        

        hence (a (+) (n *^ ( exp ( omega ,b)))) = (a (+) 0 ) by ORDINAL2: 35

        .= a by Th82

        .= (a +^ 0 ) by ORDINAL2: 27

        .= (a +^ (n *^ ( exp ( omega ,b)))) by A3, ORDINAL2: 35;

      end;

        suppose

         A4: a <> 0 & 0 in n;

        then

        consider A0 be Cantor-normal-form Ordinal-Sequence, a0 be Cantor-component Ordinal such that

         A5: ( CantorNF a) = (A0 ^ <%a0%>) by Th29;

        

         A6: ( last ( CantorNF a)) = a0 by A5, AFINSQ_1: 92;

        consider c be Ordinal, m be Nat such that

         A7: 0 in ( Segm m) & a0 = (m *^ ( exp ( omega ,c))) by ORDINAL5:def 9;

         0 in m & m in omega by A7, ORDINAL1:def 12;

        then

         A8: ( omega -exponent a0) = c by A7, ORDINAL5: 58;

        n in omega by ORDINAL1:def 12;

        then

         A9: ( omega -exponent (n *^ ( exp ( omega ,b)))) = b by A4, ORDINAL5: 58;

        then

         A10: (a0 (+) (n *^ ( exp ( omega ,b)))) = (a0 +^ (n *^ ( exp ( omega ,b)))) by A1, A4, A6, A7, A8, Th83;

        

         A11: (a (+) (n *^ ( exp ( omega ,b)))) = (( Sum^ ( CantorNF a)) (+) (n *^ ( exp ( omega ,b))))

        .= ((( Sum^ A0) +^ ( Sum^ <%a0%>)) (+) (n *^ ( exp ( omega ,b)))) by A5, Th24

        .= ((( Sum^ A0) (+) ( Sum^ <%a0%>)) (+) (n *^ ( exp ( omega ,b)))) by A5, Th84

        .= (( Sum^ A0) (+) (( Sum^ <%a0%>) (+) (n *^ ( exp ( omega ,b))))) by Th81

        .= (( Sum^ A0) (+) (a0 +^ (n *^ ( exp ( omega ,b))))) by A10, ORDINAL5: 53;

        set A = ( CantorNF a);

        per cases ;

          suppose

           A12: b = c;

          set B = (A0 ^ <%(a0 +^ (n *^ ( exp ( omega ,b))))%>);

          B is Cantor-normal-form

          proof

            

             A13: (a0 +^ (n *^ ( exp ( omega ,b)))) = ((m +^ n) *^ ( exp ( omega ,c))) by A7, A12, ORDINAL3: 46

            .= ((m + n) *^ ( exp ( omega ,c))) by CARD_2: 36;

            

             A14: 0 < m by A7, NAT_1: 44;

             A15:

            now

              let d be Ordinal;

              assume d in ( dom B);

              per cases by AFINSQ_1: 20;

                suppose

                 A16: d in ( dom A0);

                then

                 A17: (B . d) = (A0 . d) & (A0 . d) = (A . d) by A5, ORDINAL4:def 1;

                d in (( dom A0) +^ ( dom <%a0%>)) by A16, ORDINAL3: 24, TARSKI:def 3;

                then d in ( dom A) by A5, ORDINAL4:def 1;

                hence (B . d) is Cantor-component by A17, ORDINAL5:def 11;

              end;

                suppose ex k be Nat st k in ( dom <%(a0 +^ (n *^ ( exp ( omega ,b))))%>) & d = (( len A0) + k);

                then

                consider k be Nat such that

                 A18: k in ( dom <%(a0 +^ (n *^ ( exp ( omega ,b))))%>) & d = (( len A0) + k);

                k in ( Segm 1) by A18, AFINSQ_1: 33;

                then

                 A19: k = 0 by NAT_1: 44, NAT_1: 14;

                (B . d) = ( <%(a0 +^ (n *^ ( exp ( omega ,b))))%> . k) by A18, AFINSQ_1:def 3

                .= ((m + n) *^ ( exp ( omega ,c))) by A13, A19;

                hence (B . d) is Cantor-component by A14;

              end;

            end;

            now

              let d,e be Ordinal;

              assume

               A20: d in e & e in ( dom B);

              per cases by AFINSQ_1: 20;

                suppose

                 A21: e in ( dom A0);

                then

                 A22: (B . e) = (A0 . e) & (A0 . e) = (A . e) by A5, ORDINAL4:def 1;

                e in (( dom A0) +^ ( dom <%a0%>)) by A21, ORDINAL3: 24, TARSKI:def 3;

                then e in ( dom A) by A5, ORDINAL4:def 1;

                then

                 A23: ( omega -exponent (B . e)) in ( omega -exponent (A . d)) by A20, A22, ORDINAL5:def 11;

                d in ( dom A0) by A20, A21, ORDINAL1: 10;

                then (B . d) = (A0 . d) & (A0 . d) = (A . d) by A5, ORDINAL4:def 1;

                hence ( omega -exponent (B . e)) in ( omega -exponent (B . d)) by A23;

              end;

                suppose ex k be Nat st k in ( dom <%(a0 +^ (n *^ ( exp ( omega ,b))))%>) & e = (( len A0) + k);

                then

                consider k be Nat such that

                 A24: k in ( dom <%(a0 +^ (n *^ ( exp ( omega ,b))))%>) & e = (( len A0) + k);

                

                 A25: k in ( Segm 1) by A24, AFINSQ_1: 33;

                then

                 A26: k = 0 by NAT_1: 44, NAT_1: 14;

                

                 A27: (B . e) = ( <%(a0 +^ (n *^ ( exp ( omega ,b))))%> . k) by A24, AFINSQ_1:def 3

                .= ((m + n) *^ ( exp ( omega ,c))) by A13, A26;

                 0 c< (m + n) by A14, XBOOLE_1: 2, XBOOLE_0:def 8;

                then 0 in (m + n) & (m + n) in omega by ORDINAL1: 11, ORDINAL1:def 12;

                then

                 A28: ( omega -exponent (B . e)) = c by A27, ORDINAL5: 58;

                

                 A29: (A . d) = (A0 . d) & (B . d) = (A0 . d) by A5, A20, A24, A26, ORDINAL4:def 1;

                k in ( dom <%a0%>) by A25, AFINSQ_1: 33;

                then

                 A30: e in ( dom A) by A5, A24, AFINSQ_1: 23;

                ( omega -exponent (A . e)) = ( omega -exponent (B . e)) by A5, A8, A24, A26, A28, AFINSQ_1: 36;

                hence ( omega -exponent (B . e)) in ( omega -exponent (B . d)) by A20, A29, A30, ORDINAL5:def 11;

              end;

            end;

            hence thesis by A15, ORDINAL5:def 11;

          end;

          

          then (( Sum^ A0) (+) ( Sum^ <%(a0 +^ (n *^ ( exp ( omega ,b))))%>)) = (( Sum^ A0) +^ ( Sum^ <%(a0 +^ (n *^ ( exp ( omega ,b))))%>)) by Th84

          .= (( Sum^ A0) +^ (a0 +^ (n *^ ( exp ( omega ,b))))) by ORDINAL5: 53

          .= ((( Sum^ A0) +^ a0) +^ (n *^ ( exp ( omega ,b)))) by ORDINAL3: 30

          .= (( Sum^ (A0 ^ <%a0%>)) +^ (n *^ ( exp ( omega ,b)))) by ORDINAL5: 54

          .= (a +^ (n *^ ( exp ( omega ,b)))) by A5;

          hence thesis by A11, ORDINAL5: 53;

        end;

          suppose

           A31: b <> c;

          set B = (A0 ^ <%a0, (n *^ ( exp ( omega ,b)))%>);

          B is Cantor-normal-form

          proof

             A32:

            now

              let d be Ordinal;

              assume d in ( dom B);

              per cases by AFINSQ_1: 20;

                suppose

                 A33: d in ( dom A0);

                then

                 A34: (B . d) = (A0 . d) & (A0 . d) = (A . d) by A5, ORDINAL4:def 1;

                d in (( dom A0) +^ ( dom <%a0%>)) by A33, ORDINAL3: 24, TARSKI:def 3;

                then d in ( dom A) by A5, ORDINAL4:def 1;

                hence (B . d) is Cantor-component by A34, ORDINAL5:def 11;

              end;

                suppose ex k be Nat st k in ( dom <%a0, (n *^ ( exp ( omega ,b)))%>) & d = (( len A0) + k);

                then

                consider k be Nat such that

                 A35: k in ( dom <%a0, (n *^ ( exp ( omega ,b)))%>) & d = (( len A0) + k);

                k in ( Segm 2) by AFINSQ_1: 38, A35;

                per cases by NAT_1: 44, NAT_1: 23;

                  suppose

                   A36: k = 0 ;

                  (B . d) = ( <%a0, (n *^ ( exp ( omega ,b)))%> . k) by A35, AFINSQ_1:def 3

                  .= a0 by A36;

                  hence (B . d) is Cantor-component;

                end;

                  suppose

                   A37: k = 1;

                  

                   A38: (B . d) = ( <%a0, (n *^ ( exp ( omega ,b)))%> . k) by A35, AFINSQ_1:def 3

                  .= (n *^ ( exp ( omega ,b))) by A37;

                   0 <> n by A4;

                  hence (B . d) is Cantor-component by A38;

                end;

              end;

            end;

            now

              let d,e be Ordinal;

              

               A39: b in c by A1, A4, A6, A8, A31, XBOOLE_0:def 8, ORDINAL1: 11;

              assume

               A40: d in e & e in ( dom B);

              per cases by AFINSQ_1: 20;

                suppose

                 A41: e in ( dom A0);

                then

                 A42: (B . e) = (A0 . e) & (A0 . e) = (A . e) by A5, ORDINAL4:def 1;

                e in (( dom A0) +^ ( dom <%a0%>)) by A41, ORDINAL3: 24, TARSKI:def 3;

                then e in ( dom A) by A5, ORDINAL4:def 1;

                then

                 A43: ( omega -exponent (B . e)) in ( omega -exponent (A . d)) by A40, A42, ORDINAL5:def 11;

                d in ( dom A0) by A40, A41, ORDINAL1: 10;

                then (B . d) = (A0 . d) & (A0 . d) = (A . d) by A5, ORDINAL4:def 1;

                hence ( omega -exponent (B . e)) in ( omega -exponent (B . d)) by A43;

              end;

                suppose ex k2 be Nat st k2 in ( dom <%a0, (n *^ ( exp ( omega ,b)))%>) & e = (( len A0) + k2);

                then

                consider k2 be Nat such that

                 A44: k2 in ( dom <%a0, (n *^ ( exp ( omega ,b)))%>) & e = (( len A0) + k2);

                k2 in ( Segm 2) by AFINSQ_1: 38, A44;

                then

                 A45: k2 < 2 by NAT_1: 44;

                d in ( dom B) by A40, ORDINAL1: 10;

                per cases by AFINSQ_1: 20;

                  suppose

                   A46: d in ( dom A0);

                  then

                   A47: (B . d) = (A0 . d) & (A0 . d) = (A . d) by A5, ORDINAL4:def 1;

                   0 in ( Segm 1) by NAT_1: 44;

                  then 0 in ( dom <%a0%>) by AFINSQ_1: 33;

                  then (( len A0) + 0 ) in ( dom A) by A5, AFINSQ_1: 23;

                  then ( omega -exponent (A . ( len A0))) in ( omega -exponent (A . d)) by A46, ORDINAL5:def 11;

                  then

                   A48: c in ( omega -exponent (B . d)) by A5, A8, A47, AFINSQ_1: 36;

                  per cases by A45, NAT_1: 23;

                    suppose

                     A49: k2 = 0 ;

                    (B . e) = ( <%a0, (n *^ ( exp ( omega ,b)))%> . k2) by A44, AFINSQ_1:def 3

                    .= a0 by A49;

                    hence ( omega -exponent (B . e)) in ( omega -exponent (B . d)) by A8, A48;

                  end;

                    suppose

                     A50: k2 = 1;

                    (B . e) = ( <%a0, (n *^ ( exp ( omega ,b)))%> . k2) by A44, AFINSQ_1:def 3

                    .= (n *^ ( exp ( omega ,b))) by A50;

                    hence ( omega -exponent (B . e)) in ( omega -exponent (B . d)) by A9, A39, A48, ORDINAL1: 10;

                  end;

                end;

                  suppose ex k1 be Nat st k1 in ( dom <%a0, (n *^ ( exp ( omega ,b)))%>) & d = (( len A0) + k1);

                  then

                  consider k1 be Nat such that

                   A51: k1 in ( dom <%a0, (n *^ ( exp ( omega ,b)))%>) & d = (( len A0) + k1);

                  k1 in ( Segm 2) by AFINSQ_1: 38, A51;

                  then

                   A52: k1 < 2 by NAT_1: 44;

                  

                   A53: k1 = 0 & k2 = 1

                  proof

                    per cases by A45, A52, NAT_1: 23;

                      suppose k1 = 0 & k2 = 0 ;

                      hence thesis by A40, A44, A51;

                    end;

                      suppose k1 = 0 & k2 = 1;

                      hence thesis;

                    end;

                      suppose

                       A54: k1 = 1 & k2 = 0 ;

                      reconsider e, d as finite Ordinal by A44, A51;

                      e < d by A44, A51, A54, XREAL_1: 8;

                      then e in ( Segm d) by NAT_1: 44;

                      hence thesis by A40;

                    end;

                      suppose k1 = 1 & k2 = 1;

                      hence thesis by A40, A44, A51;

                    end;

                  end;

                  (B . d) = ( <%a0, (n *^ ( exp ( omega ,b)))%> . k1) by A51, AFINSQ_1:def 3

                  .= a0 by A53;

                  then

                   A55: ( omega -exponent (B . d)) = c by A8;

                  (B . e) = ( <%a0, (n *^ ( exp ( omega ,b)))%> . k2) by A44, AFINSQ_1:def 3

                  .= (n *^ ( exp ( omega ,b))) by A53;

                  hence ( omega -exponent (B . e)) in ( omega -exponent (B . d)) by A9, A39, A55;

                end;

              end;

            end;

            hence thesis by A32, ORDINAL5:def 11;

          end;

          

          then (( Sum^ A0) (+) ( Sum^ <%a0, (n *^ ( exp ( omega ,b)))%>)) = (( Sum^ A0) +^ ( Sum^ <%a0, (n *^ ( exp ( omega ,b)))%>)) by Th84

          .= (( Sum^ A0) +^ (a0 +^ (n *^ ( exp ( omega ,b))))) by Th25

          .= ((( Sum^ A0) +^ a0) +^ (n *^ ( exp ( omega ,b)))) by ORDINAL3: 30

          .= (( Sum^ (A0 ^ <%a0%>)) +^ (n *^ ( exp ( omega ,b)))) by ORDINAL5: 54

          .= (a +^ (n *^ ( exp ( omega ,b)))) by A5;

          hence thesis by A11, Th25;

        end;

      end;

    end;

    theorem :: ORDINAL7:74

    for a be Ordinal, n,m be Nat holds ((n *^ ( exp ( omega ,a))) (+) (m *^ ( exp ( omega ,a)))) = ((n + m) *^ ( exp ( omega ,a)))

    proof

      let a be Ordinal, n,m be Nat;

      per cases ;

        suppose

         A1: n = 0 ;

        

        hence ((n *^ ( exp ( omega ,a))) (+) (m *^ ( exp ( omega ,a)))) = ( 0 (+) (m *^ ( exp ( omega ,a)))) by ORDINAL2: 35

        .= ((n + m) *^ ( exp ( omega ,a))) by A1, Th82;

      end;

        suppose

         A2: n <> 0 ;

        then

         A3: 0 in n & n in omega by XBOOLE_1: 61, ORDINAL1: 11, ORDINAL1:def 12;

        ( omega -exponent ( last ( CantorNF (n *^ ( exp ( omega ,a)))))) = ( omega -exponent ( last ( {} ^ <%(n *^ ( exp ( omega ,a)))%>))) by A2, Th69

        .= ( omega -exponent (n *^ ( exp ( omega ,a)))) by AFINSQ_1: 92

        .= a by A3, ORDINAL5: 58;

        

        hence ((n *^ ( exp ( omega ,a))) (+) (m *^ ( exp ( omega ,a)))) = ((n *^ ( exp ( omega ,a))) +^ (m *^ ( exp ( omega ,a)))) by Th86

        .= ((n +^ m) *^ ( exp ( omega ,a))) by ORDINAL3: 46

        .= ((n + m) *^ ( exp ( omega ,a))) by CARD_2: 36;

      end;

    end;

    theorem :: ORDINAL7:75

    

     Th88: for a be Ordinal, n be Nat holds (a (+) n) = (a +^ n)

    proof

      let a be Ordinal, n be Nat;

      

       A1: 0 c= ( omega -exponent ( last ( CantorNF a)));

      

      thus (a (+) n) = (a (+) (n *^ 1)) by ORDINAL2: 39

      .= (a (+) (n *^ ( exp ( omega , 0 qua Ordinal)))) by ORDINAL2: 43

      .= (a +^ (n *^ ( exp ( omega , 0 qua Ordinal)))) by A1, Th86

      .= (a +^ (n *^ 1)) by ORDINAL2: 43

      .= (a +^ n) by ORDINAL2: 39;

    end;

    theorem :: ORDINAL7:76

    

     Th89: for n,m be Nat holds (n (+) m) = (n + m)

    proof

      let n,m be Nat;

      

      thus (n (+) m) = (n +^ m) by Th88

      .= (n + m) by CARD_2: 36;

    end;

    registration

      let n,m be Nat;

      identify n + m with n (+) m;

      correctness by Th89;

    end

    theorem :: ORDINAL7:77

    

     Th90: for a be Ordinal holds (a (+) 1) = ( succ a)

    proof

      let a be Ordinal;

      

      thus (a (+) 1) = (a +^ 1) by Th88

      .= ( succ a) by ORDINAL2: 31;

    end;

    theorem :: ORDINAL7:78

    for a,b be Ordinal holds (a (+) ( succ b)) = ( succ (a (+) b))

    proof

      let a,b be Ordinal;

      

      thus (a (+) ( succ b)) = (a (+) (b (+) 1)) by Th90

      .= ((a (+) b) (+) 1) by Th81

      .= ( succ (a (+) b)) by Th90;

    end;

    registration

      let a be empty Ordinal;

      cluster (a (+) a) -> empty;

      coherence ;

    end

    registration

      let a be non empty Ordinal, b be Ordinal;

      cluster (a (+) b) -> non empty;

      coherence

      proof

        assume

         A1: (a (+) b) is empty;

        set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

        set L1 = ( omega -leading_coeff ( CantorNF a));

        set L2 = ( omega -leading_coeff ( CantorNF b));

        consider C be Cantor-normal-form Ordinal-Sequence such that

         A2: (a (+) b) = ( Sum^ C) & ( rng ( omega -exponent C)) = (( rng E1) \/ ( rng E2)) and for d be object st d in ( dom C) holds (( omega -exponent (C . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C . d)) = (L1 . ((E1 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C . d)) = (L2 . ((E2 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d)))))) by Def5;

        C is empty by A1, A2;

        hence contradiction by A2;

      end;

    end

    theorem :: ORDINAL7:79

    

     Th92: for a be Ordinal holds a is limit_ordinal iff not 0 in ( rng ( omega -exponent ( CantorNF a)))

    proof

      let a be Ordinal;

      per cases ;

        suppose a = 0 ;

        hence thesis by ORDINAL2: 4;

      end;

        suppose a <> 0 ;

        then

        consider A0 be Cantor-normal-form Ordinal-Sequence, a0 be Cantor-component Ordinal such that

         A2: ( CantorNF a) = (A0 ^ <%a0%>) by Th29;

        hereby

          assume

           A3: a is limit_ordinal;

          ( omega -exponent ( last ( CantorNF a))) <> 0

          proof

            assume ( omega -exponent ( last ( CantorNF a))) = 0 ;

            then ( omega -exponent a0) = 0 by A2, AFINSQ_1: 92;

            

            then a0 = (( omega -leading_coeff a0) *^ ( exp ( omega , 0 qua Ordinal))) by Th59

            .= (( omega -leading_coeff a0) *^ 1) by ORDINAL2: 43

            .= ( omega -leading_coeff a0) by ORDINAL2: 39;

            then

             A6: ( Sum^ ( CantorNF a)) = (( Sum^ A0) +^ ( omega -leading_coeff a0)) by A2, ORDINAL5: 54;

            then

             A7: ( Sum^ A0) c= a by ORDINAL3: 24;

            ( Sum^ A0) <> a

            proof

              assume ( Sum^ A0) = a;

              then (( Sum^ A0) +^ 0 ) = (( Sum^ A0) +^ ( omega -leading_coeff a0)) by A6, ORDINAL2: 27;

              hence contradiction by ORDINAL3: 21;

            end;

            then ( Sum^ A0) in a by A7, XBOOLE_0:def 8, ORDINAL1: 11;

            then (( Sum^ A0) +^ ( omega -leading_coeff a0)) in a by A3, CARD_2: 70;

            hence contradiction by A6;

          end;

          hence not 0 in ( rng ( omega -exponent ( CantorNF a))) by Th51;

        end;

        assume

         A8: not 0 in ( rng ( omega -exponent ( CantorNF a)));

        now

          let b be Ordinal;

          assume b in a;

          then

           A9: ( succ b) in ( succ a) by ORDINAL3: 3;

           not ( succ b) = a

          proof

            assume ( succ b) = a;

            then

             A10: a = (b (+) 1) by Th90;

            set E1 = ( omega -exponent ( CantorNF b)), E2 = ( omega -exponent ( CantorNF 1));

            set L1 = ( omega -leading_coeff ( CantorNF b));

            set L2 = ( omega -leading_coeff ( CantorNF 1));

            consider C be Cantor-normal-form Ordinal-Sequence such that

             A11: (b (+) 1) = ( Sum^ C) & ( rng ( omega -exponent C)) = (( rng E1) \/ ( rng E2)) and for d be object st d in ( dom C) holds (( omega -exponent (C . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C . d)) = (L1 . ((E1 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C . d)) = (L2 . ((E2 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d)))))) by Def5;

            E2 = ( omega -exponent <%1%>) by Th71

            .= <%( omega -exponent 1)%> by Th46

            .= <% 0 %> by Th21;

            then ( rng E2) = { 0 } by AFINSQ_1: 33;

            then 0 in ( rng E2) by TARSKI:def 1;

            hence contradiction by A8, A10, A11, XBOOLE_1: 7, TARSKI:def 3;

          end;

          hence ( succ b) in a by A9, ORDINAL1: 8;

        end;

        hence thesis by ORDINAL1: 28;

      end;

    end;

    registration

      let a,b be limit_ordinal Ordinal;

      cluster (a (+) b) -> limit_ordinal;

      coherence

      proof

        set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

        set L1 = ( omega -leading_coeff ( CantorNF a));

        set L2 = ( omega -leading_coeff ( CantorNF b));

        

         A1: ( rng ( omega -exponent ( CantorNF (a (+) b)))) = (( rng E1) \/ ( rng E2)) by Th76;

         not 0 in ( rng E1) & not 0 in ( rng E2) by Th92;

        then not 0 in ( rng ( omega -exponent ( CantorNF (a (+) b)))) by A1, XBOOLE_0:def 3;

        hence thesis by Th92;

      end;

    end

    registration

      let a be Ordinal, b be non limit_ordinal Ordinal;

      cluster (a (+) b) -> non limit_ordinal;

      coherence

      proof

        set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

        set L1 = ( omega -leading_coeff ( CantorNF a));

        set L2 = ( omega -leading_coeff ( CantorNF b));

        

         A1: ( rng ( omega -exponent ( CantorNF (a (+) b)))) = (( rng E1) \/ ( rng E2)) by Th76;

         0 in ( rng E2) by Th92;

        then 0 in ( rng ( omega -exponent ( CantorNF (a (+) b)))) by A1, XBOOLE_1: 7, TARSKI:def 3;

        hence thesis by Th92;

      end;

    end

    theorem :: ORDINAL7:80

    for a,b be Ordinal, n be non zero Nat st (n *^ ( exp ( omega ,b))) c= a & a in ((n + 1) *^ ( exp ( omega ,b))) holds (( CantorNF a) . 0 ) = (n *^ ( exp ( omega ,b)))

    proof

      let a,b be Ordinal, n be non zero Nat;

      assume

       A1: (n *^ ( exp ( omega ,b))) c= a & a in ((n + 1) *^ ( exp ( omega ,b)));

      then

       A2: a <> {} ;

      then

      consider a0 be Cantor-component Ordinal, A0 be Cantor-normal-form Ordinal-Sequence such that

       A3: ( CantorNF a) = ( <%a0%> ^ A0) by ORDINAL5: 67;

      

       A4: 0 in n by XBOOLE_1: 61, ORDINAL1: 11;

      n in ( succ n) by ORDINAL1: 6;

      then 0 in ( succ n) by A4, ORDINAL1: 10;

      then

       A5: 0 in (n + 1) by Lm5;

      n in omega & (n + 1) in omega by ORDINAL1:def 12;

      then

       A7: ( omega -exponent (n *^ ( exp ( omega ,b)))) = b by A4, ORDINAL5: 58;

      ( omega -exponent ((n + 1) *^ ( exp ( omega ,b)))) = b by A5, ORDINAL5: 58;

      then b c= ( omega -exponent a) & ( omega -exponent a) c= b by A7, A1, Th22, ORDINAL1:def 2;

      

      then

       A9: b = ( omega -exponent ( Sum^ ( CantorNF a))) by XBOOLE_0:def 10

      .= ( omega -exponent (( CantorNF a) . 0 )) by Th44;

       0 in ( dom ( CantorNF a)) by A2, XBOOLE_1: 61, ORDINAL1: 11;

      then

       A10: (( CantorNF a) . 0 ) is Cantor-component by ORDINAL5:def 11;

      then

      reconsider m = ( omega -leading_coeff (( CantorNF a) . 0 )) as Nat;

      

       A11: (( CantorNF a) . 0 ) = (m *^ ( exp ( omega ,b))) by A9, A10, Th59;

      

       A12: (( CantorNF a) . 0 ) = a0 by A3, AFINSQ_1: 35;

      m = n

      proof

        assume m <> n;

        per cases by XXREAL_0: 1;

          suppose m < n;

          then (m + 1) <= n by NAT_1: 13;

          then ( Segm (m + 1)) c= ( Segm n) by NAT_1: 39;

          then ((m + 1) *^ ( exp ( omega ,b))) c= (n *^ ( exp ( omega ,b))) by ORDINAL2: 41;

          then ((m +^ 1) *^ ( exp ( omega ,b))) c= (n *^ ( exp ( omega ,b))) by CARD_2: 36;

          then

           A13: ((m *^ ( exp ( omega ,b))) +^ (1 *^ ( exp ( omega ,b)))) c= (n *^ ( exp ( omega ,b))) by ORDINAL3: 46;

          ( Sum^ A0) in ( exp ( omega ,b)) by A3, A9, A12, Th43;

          then ( Sum^ A0) in (1 *^ ( exp ( omega ,b))) by ORDINAL2: 39;

          then ((m *^ ( exp ( omega ,b))) +^ ( Sum^ A0)) in ((m *^ ( exp ( omega ,b))) +^ (1 *^ ( exp ( omega ,b)))) by ORDINAL2: 32;

          then (a0 +^ ( Sum^ A0)) in (n *^ ( exp ( omega ,b))) by A11, A12, A13;

          then ( Sum^ ( CantorNF a)) in (n *^ ( exp ( omega ,b))) by A3, ORDINAL5: 55;

          hence contradiction by A1, ORDINAL1: 12;

        end;

          suppose n < m;

          then (n + 1) <= m by NAT_1: 13;

          then ( Segm (n + 1)) c= ( Segm m) by NAT_1: 39;

          then

           A14: ((n + 1) *^ ( exp ( omega ,b))) c= (( CantorNF a) . 0 ) by A11, ORDINAL2: 41;

          (( CantorNF a) . 0 ) c= ( Sum^ ( CantorNF a)) by ORDINAL5: 56;

          then ((n + 1) *^ ( exp ( omega ,b))) c= ( Sum^ ( CantorNF a)) by A14, XBOOLE_1: 1;

          hence contradiction by A1, ORDINAL1: 12;

        end;

      end;

      hence thesis by A11;

    end;

    theorem :: ORDINAL7:81

    for a,b be Ordinal st ( rng ( omega -exponent ( CantorNF a))) = ( rng ( omega -exponent ( CantorNF b))) holds for c be Ordinal st c in ( dom ( CantorNF a)) holds (( omega -leading_coeff ( CantorNF (a (+) b))) . c) = ((( omega -leading_coeff ( CantorNF a)) . c) + (( omega -leading_coeff ( CantorNF b)) . c))

    proof

      let a,b be Ordinal;

      set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

      set L1 = ( omega -leading_coeff ( CantorNF a));

      set L2 = ( omega -leading_coeff ( CantorNF b));

      assume

       A1: ( rng E1) = ( rng E2);

      then

       A2: E1 = E2 by Th34;

      consider C be Cantor-normal-form Ordinal-Sequence such that

       A3: (a (+) b) = ( Sum^ C) & ( rng ( omega -exponent C)) = (( rng E1) \/ ( rng E2)) and

       A4: for d be object st d in ( dom C) holds (( omega -exponent (C . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C . d)) = (L1 . ((E1 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C . d)) = (L2 . ((E2 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d)))))) by Def5;

      let c be Ordinal;

      assume

       A5: c in ( dom ( CantorNF a));

      

       A6: ( dom ( CantorNF a)) = ( card ( dom E1)) by Def1

      .= ( card ( rng E1)) by CARD_1: 70

      .= ( card ( dom ( omega -exponent C))) by A1, A3, CARD_1: 70

      .= ( dom C) by Def1;

      

       A7: ( rng ( omega -exponent C)) = ( rng E1) by A1, A3;

      then

       A8: ( rng ( omega -exponent C)) = (( rng E1) /\ ( rng E2)) by A1;

      c in ( dom ( omega -exponent C)) by A5, A6, Def1;

      then (( omega -exponent C) . c) in ( rng ( omega -exponent C)) by FUNCT_1: 3;

      then

       A9: ( omega -exponent (C . c)) in (( rng E1) /\ ( rng E2)) by A5, A6, A8, Def1;

      

       A10: ( omega -exponent C) = E1 by A7, Th34;

      

       A11: c in ( dom E1) by A5, Def1;

      

      thus (( omega -leading_coeff ( CantorNF (a (+) b))) . c) = ( omega -leading_coeff (C . c)) by A3, A5, A6, Def3

      .= ((L1 . ((E1 " ) . ( omega -exponent (C . c)))) + (L2 . ((E2 " ) . ( omega -exponent (C . c))))) by A4, A5, A6, A9

      .= ((L1 . ((E1 " ) . (E1 . c))) + (L2 . ((E2 " ) . ( omega -exponent (C . c))))) by A5, A6, A10, Def1

      .= ((L1 . ((E1 " ) . (E1 . c))) + (L2 . ((E2 " ) . (E2 . c)))) by A2, A5, A6, A10, Def1

      .= ((L1 . c) + (L2 . ((E2 " ) . (E2 . c)))) by A11, FUNCT_1: 34

      .= ((L1 . c) + (L2 . c)) by A2, A11, FUNCT_1: 34;

    end;

    theorem :: ORDINAL7:82

    

     Th95: for a,b be Ordinal holds (( omega -exponent (( CantorNF (a (+) b)) . 0 )) in ( rng ( omega -exponent ( CantorNF a))) implies ( omega -exponent (( CantorNF (a (+) b)) . 0 )) = (( omega -exponent ( CantorNF a)) . 0 )) & (( omega -exponent (( CantorNF (a (+) b)) . 0 )) in ( rng ( omega -exponent ( CantorNF b))) implies ( omega -exponent (( CantorNF (a (+) b)) . 0 )) = (( omega -exponent ( CantorNF b)) . 0 ))

    proof

      let a,b be Ordinal;

      set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

      set C0 = ( CantorNF (a (+) b));

      

       A2: ( rng ( omega -exponent C0)) = (( rng E1) \/ ( rng E2)) by Th76;

      hereby

        assume ( omega -exponent (C0 . 0 )) in ( rng E1);

        then

        consider x be object such that

         A4: x in ( dom E1) & (E1 . x) = ( omega -exponent (C0 . 0 )) by FUNCT_1:def 3;

        reconsider x as Ordinal by A4;

        x = 0

        proof

          assume

           A5: x <> 0 ;

          then 0 c< x by XBOOLE_1: 2, XBOOLE_0:def 8;

          then

           A6: 0 in x by ORDINAL1: 11;

          then

           A7: 0 in ( dom E1) by A4, ORDINAL1: 10;

          then (E1 . 0 ) in ( rng E1) by FUNCT_1: 3;

          then (E1 . 0 ) in ( rng ( omega -exponent C0)) by A2, XBOOLE_1: 7, TARSKI:def 3;

          then

          consider y be object such that

           A8: y in ( dom ( omega -exponent C0)) & (( omega -exponent C0) . y) = (E1 . 0 ) by FUNCT_1:def 3;

          reconsider y as Ordinal by A8;

          

           A9: y in ( dom C0) by A8, Def1;

          then

           A10: ( omega -exponent (C0 . y)) = (E1 . 0 ) by A8, Def1;

          per cases ;

            suppose y = 0 ;

            hence contradiction by A4, A5, A7, A10, FUNCT_1:def 4;

          end;

            suppose y <> 0 ;

            then 0 c< y by XBOOLE_1: 2, XBOOLE_0:def 8;

            then 0 in y by ORDINAL1: 11;

            then

             A11: (E1 . 0 ) in (E1 . x) by A4, A9, A10, ORDINAL5:def 11;

            

             A12: x in ( dom ( CantorNF a)) by A4, Def1;

            then ( omega -exponent (( CantorNF a) . 0 )) in (E1 . x) by A11, Def1, A6, ORDINAL1: 10;

            then ( omega -exponent (( CantorNF a) . 0 )) in ( omega -exponent (( CantorNF a) . x)) by A12, Def1;

            hence contradiction by A6, A12, ORDINAL5:def 11;

          end;

        end;

        hence ( omega -exponent (C0 . 0 )) = (E1 . 0 ) by A4;

      end;

      assume ( omega -exponent (C0 . 0 )) in ( rng E2);

      then

      consider x be object such that

       A14: x in ( dom E2) & (E2 . x) = ( omega -exponent (C0 . 0 )) by FUNCT_1:def 3;

      reconsider x as Ordinal by A14;

      x = 0

      proof

        assume

         A15: x <> 0 ;

        then 0 c< x by XBOOLE_1: 2, XBOOLE_0:def 8;

        then

         A16: 0 in x by ORDINAL1: 11;

        then

         A17: 0 in ( dom E2) by A14, ORDINAL1: 10;

        then (E2 . 0 ) in ( rng E2) by FUNCT_1: 3;

        then (E2 . 0 ) in ( rng ( omega -exponent C0)) by A2, XBOOLE_1: 7, TARSKI:def 3;

        then

        consider y be object such that

         A18: y in ( dom ( omega -exponent C0)) & (( omega -exponent C0) . y) = (E2 . 0 ) by FUNCT_1:def 3;

        reconsider y as Ordinal by A18;

        

         A19: y in ( dom C0) by A18, Def1;

        then

         A20: ( omega -exponent (C0 . y)) = (E2 . 0 ) by A18, Def1;

        per cases ;

          suppose y = 0 ;

          hence contradiction by A14, A15, A17, A20, FUNCT_1:def 4;

        end;

          suppose y <> 0 ;

          then 0 c< y by XBOOLE_1: 2, XBOOLE_0:def 8;

          then 0 in y by ORDINAL1: 11;

          then

           A21: (E2 . 0 ) in (E2 . x) by A14, A19, A20, ORDINAL5:def 11;

          

           A22: x in ( dom ( CantorNF b)) by A14, Def1;

          then ( omega -exponent (( CantorNF b) . 0 )) in (E2 . x) by A21, Def1, A16, ORDINAL1: 10;

          then ( omega -exponent (( CantorNF b) . 0 )) in ( omega -exponent (( CantorNF b) . x)) by A22, Def1;

          hence contradiction by A16, A22, ORDINAL5:def 11;

        end;

      end;

      hence ( omega -exponent (C0 . 0 )) = (E2 . 0 ) by A14;

    end;

    theorem :: ORDINAL7:83

    for a,b be Ordinal holds (( omega -exponent (( CantorNF (a (+) b)) . 0 )) in (( rng ( omega -exponent ( CantorNF a))) \ ( rng ( omega -exponent ( CantorNF b)))) implies (( CantorNF (a (+) b)) . 0 ) = (( CantorNF a) . 0 )) & (( omega -exponent (( CantorNF (a (+) b)) . 0 )) in (( rng ( omega -exponent ( CantorNF b))) \ ( rng ( omega -exponent ( CantorNF a)))) implies (( CantorNF (a (+) b)) . 0 ) = (( CantorNF b) . 0 )) & (( omega -exponent (( CantorNF (a (+) b)) . 0 )) in (( rng ( omega -exponent ( CantorNF a))) /\ ( rng ( omega -exponent ( CantorNF b)))) implies (( CantorNF (a (+) b)) . 0 ) = ((( CantorNF a) . 0 ) +^ (( CantorNF b) . 0 )))

    proof

      let a,b be Ordinal;

      set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

      set L1 = ( omega -leading_coeff ( CantorNF a));

      set L2 = ( omega -leading_coeff ( CantorNF b));

      set C0 = ( CantorNF (a (+) b));

      per cases ;

        suppose

         A1: (a (+) b) <> {} ;

        then 0 c< ( dom C0) by XBOOLE_1: 2, XBOOLE_0:def 8;

        then

         A2: 0 in ( dom C0) by ORDINAL1: 11;

        consider C be Cantor-normal-form Ordinal-Sequence such that

         A3: (a (+) b) = ( Sum^ C) & ( rng ( omega -exponent C)) = (( rng E1) \/ ( rng E2)) and

         A4: for d be object st d in ( dom C) holds (( omega -exponent (C . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C . d)) = (L1 . ((E1 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C . d)) = (L2 . ((E2 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d)))))) by Def5;

        C <> {} by A1, A3, ORDINAL5: 52;

        then 0 c< ( dom C) by XBOOLE_1: 2, XBOOLE_0:def 8;

        then

         A5: 0 in ( dom C) by ORDINAL1: 11;

        hereby

          assume

           A6: ( omega -exponent (C0 . 0 )) in (( rng E1) \ ( rng E2));

          then E1 <> {} ;

          then 0 c< ( dom E1) by XBOOLE_1: 2, XBOOLE_0:def 8;

          then

           A7: 0 in ( dom E1) by ORDINAL1: 11;

          

           A8: ( omega -leading_coeff (C . 0 )) = (L1 . ((E1 " ) . ( omega -exponent (C . 0 )))) by A3, A4, A5, A6

          .= (L1 . ((E1 " ) . (E1 . 0 ))) by A3, A6, Th95

          .= (L1 . 0 ) by A7, FUNCT_1: 34;

          

           A9: 0 in ( dom ( CantorNF a)) by A7, Def1;

          

          thus (C0 . 0 ) = (( omega -leading_coeff (C0 . 0 )) *^ ( exp ( omega ,( omega -exponent (C0 . 0 ))))) by A2, Th64

          .= ((L1 . 0 ) *^ ( exp ( omega ,(E1 . 0 )))) by A3, A6, A8, Th95

          .= (( CantorNF a) . 0 ) by A9, Th65;

        end;

        hereby

          assume

           A10: ( omega -exponent (C0 . 0 )) in (( rng E2) \ ( rng E1));

          then E2 <> {} ;

          then 0 c< ( dom E2) by XBOOLE_1: 2, XBOOLE_0:def 8;

          then

           A11: 0 in ( dom E2) by ORDINAL1: 11;

          

           A12: ( omega -leading_coeff (C . 0 )) = (L2 . ((E2 " ) . ( omega -exponent (C . 0 )))) by A3, A4, A5, A10

          .= (L2 . ((E2 " ) . (E2 . 0 ))) by A3, A10, Th95

          .= (L2 . 0 ) by A11, FUNCT_1: 34;

          

           A13: 0 in ( dom ( CantorNF b)) by A11, Def1;

          

          thus (C0 . 0 ) = (( omega -leading_coeff (C0 . 0 )) *^ ( exp ( omega ,( omega -exponent (C0 . 0 ))))) by A2, Th64

          .= ((L2 . 0 ) *^ ( exp ( omega ,(E2 . 0 )))) by A3, A10, A12, Th95

          .= (( CantorNF b) . 0 ) by A13, Th65;

        end;

        assume

         A14: ( omega -exponent (C0 . 0 )) in (( rng E1) /\ ( rng E2));

        then

         A15: ( omega -exponent (C0 . 0 )) in ( rng E1) & ( omega -exponent (C0 . 0 )) in ( rng E2) by XBOOLE_0:def 4;

        then E1 <> {} & E2 <> {} ;

        then 0 c< ( dom E1) & 0 c< ( dom E2) by XBOOLE_1: 2, XBOOLE_0:def 8;

        then

         A16: 0 in ( dom E1) & 0 in ( dom E2) by ORDINAL1: 11;

        

         A17: ( omega -leading_coeff (C . 0 )) = ((L1 . ((E1 " ) . ( omega -exponent (C . 0 )))) + (L2 . ((E2 " ) . ( omega -exponent (C . 0 ))))) by A3, A4, A5, A14

        .= ((L1 . ((E1 " ) . (E1 . 0 ))) + (L2 . ((E2 " ) . ( omega -exponent (C0 . 0 ))))) by A3, A15, Th95

        .= ((L1 . ((E1 " ) . (E1 . 0 ))) + (L2 . ((E2 " ) . (E2 . 0 )))) by A15, Th95

        .= ((L1 . 0 ) + (L2 . ((E2 " ) . (E2 . 0 )))) by A16, FUNCT_1: 34

        .= ((L1 . 0 ) + (L2 . 0 )) by A16, FUNCT_1: 34;

        

         A18: 0 in ( dom ( CantorNF a)) & 0 in ( dom ( CantorNF b)) by A16, Def1;

        

        thus (C0 . 0 ) = (( omega -leading_coeff (C0 . 0 )) *^ ( exp ( omega ,( omega -exponent (C0 . 0 ))))) by A2, Th64

        .= (((L1 . 0 ) +^ (L2 . 0 )) *^ ( exp ( omega ,( omega -exponent (C0 . 0 ))))) by A3, A17, CARD_2: 36

        .= (((L1 . 0 ) *^ ( exp ( omega ,( omega -exponent (C0 . 0 ))))) +^ ((L2 . 0 ) *^ ( exp ( omega ,( omega -exponent (C0 . 0 )))))) by ORDINAL3: 46

        .= (((L1 . 0 ) *^ ( exp ( omega ,(E1 . 0 )))) +^ ((L2 . 0 ) *^ ( exp ( omega ,( omega -exponent (C0 . 0 )))))) by A15, Th95

        .= (((L1 . 0 ) *^ ( exp ( omega ,(E1 . 0 )))) +^ ((L2 . 0 ) *^ ( exp ( omega ,(E2 . 0 ))))) by A15, Th95

        .= ((( CantorNF a) . 0 ) +^ ((L2 . 0 ) *^ ( exp ( omega ,(E2 . 0 ))))) by A18, Th65

        .= ((( CantorNF a) . 0 ) +^ (( CantorNF b) . 0 )) by A18, Th65;

      end;

        suppose (a (+) b) = {} ;

        then a = {} & b = {} ;

        hence thesis;

      end;

    end;

    theorem :: ORDINAL7:84

    

     Th97: for a,b be Ordinal, x be object holds (( omega -exponent ( CantorNF a)) . x) c= (( omega -exponent ( CantorNF (a (+) b))) . x)

    proof

      let a,b be Ordinal, x be object;

      set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

      set L1 = ( omega -leading_coeff ( CantorNF a));

      set L2 = ( omega -leading_coeff ( CantorNF b));

      set C0 = ( CantorNF (a (+) b));

      assume not (E1 . x) c= (( omega -exponent C0) . x);

      then

       A1: (( omega -exponent C0) . x) in (E1 . x) by ORDINAL1: 16;

      then x in ( dom E1) by FUNCT_1:def 2;

      then

      reconsider x as Ordinal;

      defpred P[ Ordinal] means (( omega -exponent C0) . $1) in (E1 . $1);

      

       A2: ex z be Ordinal st P[z]

      proof

        take x;

        thus thesis by A1;

      end;

      consider y be Ordinal such that

       A3: P[y] & for z be Ordinal st P[z] holds y c= z from ORDINAL1:sch 1( A2);

      

       A4: ( rng ( omega -exponent C0)) = (( rng E1) \/ ( rng E2)) by Th76;

      

       A5: y in ( dom E1) by A3, FUNCT_1:def 2;

      then (E1 . y) in ( rng E1) by FUNCT_1: 3;

      then (E1 . y) in ( rng ( omega -exponent C0)) by A4, XBOOLE_1: 7, TARSKI:def 3;

      then

      consider z be object such that

       A6: z in ( dom ( omega -exponent C0)) & (( omega -exponent C0) . z) = (E1 . y) by FUNCT_1:def 3;

      reconsider z as Ordinal by A6;

      

       A7: z in ( dom C0) by A6, Def1;

      

       A8: z in y

      proof

        assume not z in y;

        per cases by ORDINAL1: 14;

          suppose z = y;

          hence contradiction by A3, A6;

        end;

          suppose

           A9: y in z;

          then ( omega -exponent (C0 . z)) in ( omega -exponent (C0 . y)) by A7, ORDINAL5:def 11;

          then (E1 . y) in ( omega -exponent (C0 . y)) by A6, A7, Def1;

          hence contradiction by A3, A7, A9, Def1, ORDINAL1: 10;

        end;

      end;

      

       A10: y in ( dom ( CantorNF a)) by A5, Def1;

      then ( omega -exponent (( CantorNF a) . y)) in ( omega -exponent (( CantorNF a) . z)) by A8, ORDINAL5:def 11;

      then (E1 . y) in ( omega -exponent (( CantorNF a) . z)) by A10, Def1;

      then (E1 . y) in (E1 . z) by A8, A10, Def1, ORDINAL1: 10;

      then y c= z by A3, A6;

      then z in z by A8;

      hence contradiction;

    end;

    theorem :: ORDINAL7:85

    

     Th98: for a,b be Ordinal, x be object holds (( CantorNF a) . x) c= (( CantorNF (a (+) b)) . x)

    proof

      let a,b be Ordinal, x be object;

      set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

      set L1 = ( omega -leading_coeff ( CantorNF a));

      set L2 = ( omega -leading_coeff ( CantorNF b));

      set C0 = ( CantorNF (a (+) b));

      consider C be Cantor-normal-form Ordinal-Sequence such that

       A1: (a (+) b) = ( Sum^ C) & ( rng ( omega -exponent C)) = (( rng E1) \/ ( rng E2)) and

       A2: for d be object st d in ( dom C) holds (( omega -exponent (C . d)) in (( rng E1) \ ( rng E2)) implies ( omega -leading_coeff (C . d)) = (L1 . ((E1 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E2) \ ( rng E1)) implies ( omega -leading_coeff (C . d)) = (L2 . ((E2 " ) . ( omega -exponent (C . d))))) & (( omega -exponent (C . d)) in (( rng E1) /\ ( rng E2)) implies ( omega -leading_coeff (C . d)) = ((L1 . ((E1 " ) . ( omega -exponent (C . d)))) + (L2 . ((E2 " ) . ( omega -exponent (C . d)))))) by Def5;

      assume not (( CantorNF a) . x) c= (C0 . x);

      then

       A3: (C0 . x) in (( CantorNF a) . x) by ORDINAL1: 16;

      then

       A4: x in ( dom ( CantorNF a)) by FUNCT_1:def 2;

      then

      reconsider x as Ordinal;

      ( dom ( CantorNF a)) c= ( dom ( CantorNF (a (+) b))) by Th77;

      then

       A5: x in ( dom C0) by A4;

      then

       A6: (C0 . x) = ((( omega -leading_coeff C0) . x) *^ ( exp ( omega ,(( omega -exponent C0) . x)))) by Th65;

      

       A7: (( CantorNF a) . x) = ((L1 . x) *^ ( exp ( omega ,(E1 . x)))) by A4, Th65;

      

       A8: (E1 . x) = (( omega -exponent C0) . x)

      proof

        

         A9: (E1 . x) c= (( omega -exponent C0) . x) by Th97;

        assume (E1 . x) <> (( omega -exponent C0) . x);

        then (E1 . x) in (( omega -exponent C0) . x) by A9, XBOOLE_0:def 8, ORDINAL1: 11;

        then ( exp ( omega ,(E1 . x))) in ( exp ( omega ,(( omega -exponent C0) . x))) by ORDINAL4: 24;

        then

         A10: (( CantorNF a) . x) in ( exp ( omega ,(( omega -exponent C0) . x))) by A7, Th42;

        x in ( dom ( omega -leading_coeff C0)) by A5, Def3;

        then (( omega -leading_coeff C0) . x) <> {} by FUNCT_1:def 9;

        then 0 c< (( omega -leading_coeff C0) . x) by XBOOLE_1: 2, XBOOLE_0:def 8;

        then 0 in (( omega -leading_coeff C0) . x) by ORDINAL1: 11;

        then (1 *^ ( exp ( omega ,(( omega -exponent C0) . x)))) c= (C0 . x) by A6, CARD_1: 49, ZFMISC_1: 31, ORDINAL2: 41;

        then ( exp ( omega ,(( omega -exponent C0) . x))) c= (C0 . x) by ORDINAL2: 39;

        hence contradiction by A3, A10;

      end;

      then (( omega -leading_coeff C0) . x) in (L1 . x) by A3, A6, A7, ORDINAL3: 34;

      then

       A11: ( omega -leading_coeff (C0 . x)) in (L1 . x) by A5, Def3;

      

       A12: x in ( dom E1) by A4, Def1;

      then (( omega -exponent C0) . x) in ( rng E1) by A8, FUNCT_1: 3;

      then

       A13: ( omega -exponent (C0 . x)) in ( rng E1) by A5, Def1;

      per cases ;

        suppose ( omega -exponent (C0 . x)) in ( rng E2);

        then ( omega -exponent (C0 . x)) in (( rng E1) /\ ( rng E2)) by A13, XBOOLE_0:def 4;

        

        then ( omega -leading_coeff (C0 . x)) = ((L1 . ((E1 " ) . ( omega -exponent (C0 . x)))) + (L2 . ((E2 " ) . ( omega -exponent (C0 . x))))) by A1, A2, A5

        .= ((L1 . ((E1 " ) . (E1 . x))) + (L2 . ((E2 " ) . ( omega -exponent (C0 . x))))) by A5, A8, Def1

        .= ((L1 . x) + (L2 . ((E2 " ) . ( omega -exponent (C0 . x))))) by A12, FUNCT_1: 34;

        then ( Segm (L1 . x)) c= ( Segm ( omega -leading_coeff (C0 . x))) by NAT_1: 11, NAT_1: 39;

        then ( omega -leading_coeff (C0 . x)) in ( omega -leading_coeff (C0 . x)) by A11;

        hence contradiction;

      end;

        suppose not ( omega -exponent (C0 . x)) in ( rng E2);

        then ( omega -exponent (C0 . x)) in (( rng E1) \ ( rng E2)) by A13, XBOOLE_0:def 5;

        

        then ( omega -leading_coeff (C0 . x)) = (L1 . ((E1 " ) . ( omega -exponent (C0 . x)))) by A1, A2, A5

        .= (L1 . ((E1 " ) . (E1 . x))) by A5, A8, Def1

        .= (L1 . x) by A12, FUNCT_1: 34;

        hence contradiction by A11;

      end;

    end;

    theorem :: ORDINAL7:86

    

     Th99: for a,b be Ordinal holds a c= (a (+) b)

    proof

      let a,b be Ordinal;

      set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

      

       A1: ( dom ( CantorNF a)) c= ( dom ( CantorNF (a (+) b))) by Th77;

      for x be object st x in ( dom ( CantorNF a)) holds (( CantorNF a) . x) c= (( CantorNF (a (+) b)) . x) by Th98;

      then ( Sum^ ( CantorNF a)) c= ( Sum^ ( CantorNF (a (+) b))) by A1, Th28;

      hence thesis;

    end;

    theorem :: ORDINAL7:87

    

     Th100: for a,b be Ordinal holds ( omega -exponent (a (+) b)) = (( omega -exponent a) \/ ( omega -exponent b))

    proof

      let a,b be Ordinal;

      per cases ;

        suppose

         A1: (a (+) b) <> {} ;

        ( omega -exponent a) c= ( omega -exponent (a (+) b)) & ( omega -exponent b) c= ( omega -exponent (a (+) b)) by Th22, Th99;

        then

         A2: (( omega -exponent a) \/ ( omega -exponent b)) c= ( omega -exponent (a (+) b)) by XBOOLE_1: 8;

        set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

        set C0 = ( CantorNF (a (+) b));

         0 c< ( dom C0) by A1, XBOOLE_1: 2, XBOOLE_0:def 8;

        then

         A3: 0 in ( dom C0) by ORDINAL1: 11;

        then 0 in ( dom ( omega -exponent C0)) by Def1;

        then (( omega -exponent C0) . 0 ) in ( rng ( omega -exponent C0)) by FUNCT_1: 3;

        then (( omega -exponent C0) . 0 ) in (( rng E1) \/ ( rng E2)) by Th76;

        per cases by XBOOLE_0:def 3;

          suppose

           A4: (( omega -exponent C0) . 0 ) in ( rng E1);

          then ( omega -exponent (C0 . 0 )) in ( rng E1) by A3, Def1;

          then

           A5: ( omega -exponent (C0 . 0 )) = (E1 . 0 ) by Th95;

          E1 <> {} by A4;

          then 0 c< ( dom E1) by XBOOLE_1: 2, XBOOLE_0:def 8;

          then 0 in ( dom E1) by ORDINAL1: 11;

          then

           A6: 0 in ( dom ( CantorNF a)) by Def1;

          ( omega -exponent (a (+) b)) = ( omega -exponent ( Sum^ C0))

          .= ( omega -exponent (C0 . 0 )) by Th44

          .= ( omega -exponent (( CantorNF a) . 0 )) by A5, A6, Def1

          .= ( omega -exponent ( Sum^ ( CantorNF a))) by Th44

          .= ( omega -exponent a);

          then ( omega -exponent (a (+) b)) c= (( omega -exponent a) \/ ( omega -exponent b)) by XBOOLE_1: 7;

          hence thesis by A2, XBOOLE_0:def 10;

        end;

          suppose

           A7: (( omega -exponent C0) . 0 ) in ( rng E2);

          then ( omega -exponent (C0 . 0 )) in ( rng E2) by A3, Def1;

          then

           A8: ( omega -exponent (C0 . 0 )) = (E2 . 0 ) by Th95;

          E2 <> {} by A7;

          then 0 c< ( dom E2) by XBOOLE_1: 2, XBOOLE_0:def 8;

          then 0 in ( dom E2) by ORDINAL1: 11;

          then

           A9: 0 in ( dom ( CantorNF b)) by Def1;

          ( omega -exponent (a (+) b)) = ( omega -exponent ( Sum^ C0))

          .= ( omega -exponent (C0 . 0 )) by Th44

          .= ( omega -exponent (( CantorNF b) . 0 )) by A8, A9, Def1

          .= ( omega -exponent ( Sum^ ( CantorNF b))) by Th44

          .= ( omega -exponent b);

          then ( omega -exponent (a (+) b)) c= (( omega -exponent a) \/ ( omega -exponent b)) by XBOOLE_1: 7;

          hence thesis by A2, XBOOLE_0:def 10;

        end;

      end;

        suppose (a (+) b) = {} ;

        then a = 0 & b = 0 ;

        hence thesis;

      end;

    end;

    theorem :: ORDINAL7:88

    

     Th101: for a,b,c be Ordinal st a in ( exp ( omega ,c)) & b in ( exp ( omega ,c)) holds (a (+) b) in ( exp ( omega ,c))

    proof

      let a,b,c be Ordinal;

      assume

       A1: a in ( exp ( omega ,c)) & b in ( exp ( omega ,c));

      per cases ;

        suppose a = 0 ;

        hence thesis by A1, Th82;

      end;

        suppose b = 0 ;

        hence thesis by A1, Th82;

      end;

        suppose

         A2: a <> {} & b <> {} ;

        then 0 c< a & 0 c< b by XBOOLE_1: 2, XBOOLE_0:def 8;

        then 0 in a & 0 in b by ORDINAL1: 11;

        then ( omega -exponent a) in c & ( omega -exponent b) in c by A1, Th23;

        then (( omega -exponent a) \/ ( omega -exponent b)) in c by ORDINAL3: 12;

        then

         A3: ( omega -exponent (a (+) b)) in c by Th100;

        

         A4: not c c= ( omega -exponent (a (+) b))

        proof

          assume c c= ( omega -exponent (a (+) b));

          then ( omega -exponent (a (+) b)) in ( omega -exponent (a (+) b)) by A3;

          hence contradiction;

        end;

         0 c< (a (+) b) by A2, XBOOLE_1: 2, XBOOLE_0:def 8;

        then 0 in (a (+) b) by ORDINAL1: 11;

        then not ( exp ( omega ,c)) c= (a (+) b) by A4, ORDINAL5:def 10;

        hence thesis by ORDINAL1: 16;

      end;

    end;

    

     Lm9: for a be Ordinal, n be Nat holds ((n *^ ( exp ( omega ,a))) +^ ( exp ( omega ,a))) = ((n *^ ( exp ( omega ,a))) (+) ( exp ( omega ,a)))

    proof

      let a be Ordinal, n be Nat;

      

       A1: ( exp ( omega ,a)) = (1 *^ ( exp ( omega ,a))) by ORDINAL2: 39;

       0 in 1 & 1 in omega by CARD_1: 49, TARSKI:def 1;

      then ( omega -exponent ( exp ( omega ,a))) = a by A1, ORDINAL5: 58;

      hence thesis by Th83;

    end;

    scheme :: ORDINAL7:sch1

    OrdinalCNFIndA { P[ non empty Ordinal] } :

for a be non empty Ordinal holds P[a]

      provided

       A1: for a be Ordinal, n be non zero Nat holds P[(n *^ ( exp ( omega ,a)))]

       and

       A2: for a be Ordinal, b be non empty Ordinal, n be non zero Nat st P[b] & not a in ( rng ( omega -exponent ( CantorNF b))) holds P[(b (+) (n *^ ( exp ( omega ,a))))];

      defpred R[ Nat] means for a be non empty Ordinal st $1 = ( len ( CantorNF a)) holds P[a];

      

       A3: R[1]

      proof

        let a be non empty Ordinal;

        assume

         A4: 1 = ( len ( CantorNF a));

        then 0 in ( dom ( CantorNF a)) by CARD_1: 49, TARSKI:def 1;

        then (( CantorNF a) . 0 ) is Cantor-component by ORDINAL5:def 11;

        then

        consider b be Ordinal, n be Nat such that

         A5: 0 in ( Segm n) & (( CantorNF a) . 0 ) = (n *^ ( exp ( omega ,b))) by ORDINAL5:def 9;

        

         A6: n is non zero by A5;

        ( CantorNF a) = <%(( CantorNF a) . 0 )%> by A4, AFINSQ_1: 34;

        then ( Sum^ ( CantorNF a)) = (n *^ ( exp ( omega ,b))) by A5, ORDINAL5: 53;

        hence thesis by A1, A6;

      end;

      

       A7: for k be non zero Nat st R[k] holds R[(k + 1)]

      proof

        let k be non zero Nat;

        assume

         A8: R[k];

        let a be non empty Ordinal;

        assume

         A9: (k + 1) = ( len ( CantorNF a));

        consider c be Cantor-component Ordinal, A0 be Cantor-normal-form Ordinal-Sequence such that

         A10: ( CantorNF a) = ( <%c%> ^ A0) by ORDINAL5: 67;

        consider b be Ordinal, n be Nat such that

         A11: 0 in ( Segm n) & c = (n *^ ( exp ( omega ,b))) by ORDINAL5:def 9;

        reconsider n as non zero Nat by A11;

        (k + 1) = (( len <%c%>) + ( len A0)) by A9, A10, AFINSQ_1: 17

        .= (1 + ( len A0)) by AFINSQ_1: 34;

        then

         A12: ( len A0) = k;

        then A0 <> {} ;

        then

        reconsider a0 = ( Sum^ A0) as non empty Ordinal;

         not b in ( rng ( omega -exponent ( CantorNF a0)))

        proof

           0 in n & n in omega by A11, ORDINAL1:def 12;

          then ( omega -exponent c) = b by A11, ORDINAL5: 58;

          then

           A13: ( Sum^ A0) in ( exp ( omega ,b)) by A10, Th43;

          A0 <> {} by A12;

          then 0 c< ( Sum^ A0) by XBOOLE_1: 2, XBOOLE_0:def 8;

          then 0 in ( Sum^ A0) by ORDINAL1: 11;

          then ( omega -exponent ( Sum^ A0)) in b by A13, Th23;

          then

           A14: ( omega -exponent (A0 . 0 )) in b by Th44;

          assume b in ( rng ( omega -exponent ( CantorNF a0)));

          then

          consider x be object such that

           A15: x in ( dom ( omega -exponent A0)) & (( omega -exponent A0) . x) = b by FUNCT_1:def 3;

          reconsider x as Ordinal by A15;

          

           A16: x in ( dom A0) by A15, Def1;

          then

           A17: b = ( omega -exponent (A0 . x)) by A15, Def1;

          per cases ;

            suppose x = 0 ;

            hence contradiction by A14, A17;

          end;

            suppose x <> 0 ;

            then 0 c< x by XBOOLE_1: 2, XBOOLE_0:def 8;

            then 0 in x by ORDINAL1: 11;

            hence contradiction by A14, A16, A17, ORDINAL5:def 11;

          end;

        end;

        then P[(a0 (+) (n *^ ( exp ( omega ,b))))] by A2, A8, A12;

        then P[(( Sum^ <%c%>) (+) a0)] by A11, ORDINAL5: 53;

        then P[(( Sum^ <%c%>) +^ a0)] by A10, Th84;

        then P[(c +^ a0)] by ORDINAL5: 53;

        then P[( Sum^ ( <%c%> ^ A0))] by ORDINAL5: 55;

        hence thesis by A10;

      end;

      

       A18: for k be non zero Nat holds R[k] from NAT_1:sch 10( A3, A7);

      let a be non empty Ordinal;

      ( len ( CantorNF a)) is non zero;

      hence thesis by A18;

    end;

    scheme :: ORDINAL7:sch2

    OrdinalCNFIndB { P[ non empty Ordinal] } :

for a be non empty Ordinal holds P[a]

      provided

       A1: for a be Ordinal holds P[( exp ( omega ,a))]

       and

       A2: for a be Ordinal, n be non zero Nat st P[(n *^ ( exp ( omega ,a)))] holds P[((n + 1) *^ ( exp ( omega ,a)))]

       and

       A3: for a be Ordinal, b be non empty Ordinal, n be non zero Nat st P[b] & not a in ( rng ( omega -exponent ( CantorNF b))) holds P[(b (+) (n *^ ( exp ( omega ,a))))];

      defpred Q[ Nat] means for a be Ordinal, n be non zero Nat st $1 = n holds P[(n *^ ( exp ( omega ,a)))];

      

       A4: Q[1]

      proof

        let a be Ordinal, n be non zero Nat;

        assume 1 = n;

        then (n *^ ( exp ( omega ,a))) = ( exp ( omega ,a)) by ORDINAL2: 39;

        hence thesis by A1;

      end;

      

       A5: for k be non zero Nat st Q[k] holds Q[(k + 1)]

      proof

        let k be non zero Nat;

        assume

         A6: Q[k];

        let a be Ordinal, n be non zero Nat;

        assume

         A7: (k + 1) = n;

        P[(k *^ ( exp ( omega ,a)))] by A6;

        hence thesis by A2, A7;

      end;

      for k be non zero Nat holds Q[k] from NAT_1:sch 10( A4, A5);

      then

       A8: for a be Ordinal, n be non zero Nat holds P[(n *^ ( exp ( omega ,a)))];

      for a be non empty Ordinal holds P[a] from OrdinalCNFIndA( A8, A3);

      hence thesis;

    end;

    scheme :: ORDINAL7:sch3

    OrdinalCNFIndC { P[ non empty Ordinal] } :

for a be non empty Ordinal holds P[a]

      provided

       A1: for a be Ordinal holds P[( exp ( omega ,a))]

       and

       A2: for a be Ordinal, b be non empty Ordinal st P[b] holds P[(b (+) ( exp ( omega ,a)))];

      defpred Q[ Nat] means for a be Ordinal, n be non zero Nat st $1 = n holds P[(n *^ ( exp ( omega ,a)))];

      

       A3: Q[1]

      proof

        let a be Ordinal, n be non zero Nat;

        assume 1 = n;

        then (n *^ ( exp ( omega ,a))) = ( exp ( omega ,a)) by ORDINAL2: 39;

        hence thesis by A1;

      end;

      

       A4: for k be non zero Nat st Q[k] holds Q[(k + 1)]

      proof

        let k be non zero Nat;

        assume

         A5: Q[k];

        let a be Ordinal, n be non zero Nat;

        assume (k + 1) = n;

        

        then (n *^ ( exp ( omega ,a))) = (( Segm (k + 1)) *^ ( exp ( omega ,a)))

        .= (( succ ( Segm k)) *^ ( exp ( omega ,a))) by NAT_1: 38

        .= ((k *^ ( exp ( omega ,a))) +^ ( exp ( omega ,a))) by ORDINAL2: 36

        .= ((k *^ ( exp ( omega ,a))) (+) ( exp ( omega ,a))) by Lm9;

        hence thesis by A2, A5;

      end;

      

       A7: for k be non zero Nat holds Q[k] from NAT_1:sch 10( A3, A4);

      defpred R[ Nat] means for a be non empty Ordinal st $1 = ( len ( CantorNF a)) holds P[a];

      

       A8: R[1]

      proof

        let a be non empty Ordinal;

        assume

         A9: 1 = ( len ( CantorNF a));

        then 0 in ( dom ( CantorNF a)) by CARD_1: 49, TARSKI:def 1;

        then (( CantorNF a) . 0 ) is Cantor-component by ORDINAL5:def 11;

        then

        consider b be Ordinal, n be Nat such that

         A10: 0 in ( Segm n) & (( CantorNF a) . 0 ) = (n *^ ( exp ( omega ,b))) by ORDINAL5:def 9;

        

         A11: n is non zero by A10;

        ( CantorNF a) = <%(( CantorNF a) . 0 )%> by A9, AFINSQ_1: 34;

        then ( Sum^ ( CantorNF a)) = (n *^ ( exp ( omega ,b))) by A10, ORDINAL5: 53;

        hence thesis by A7, A11;

      end;

      defpred S[ Nat] means for a be Ordinal, b be non empty Ordinal, n be non zero Nat st $1 = n & P[b] holds P[(b (+) (n *^ ( exp ( omega ,a))))];

      

       A12: S[1]

      proof

        let a be Ordinal, b be non empty Ordinal, n be non zero Nat;

        assume

         A13: 1 = n & P[b];

        then P[(b (+) ( exp ( omega ,a)))] by A2;

        hence thesis by A13, ORDINAL2: 39;

      end;

      

       A14: for k be non zero Nat st S[k] holds S[(k + 1)]

      proof

        let k be non zero Nat;

        assume

         A15: S[k];

        let a be Ordinal, b be non empty Ordinal, n be non zero Nat;

        assume

         A16: (k + 1) = n & P[b];

        then P[(b (+) (k *^ ( exp ( omega ,a))))] by A15;

        then P[((b (+) (k *^ ( exp ( omega ,a)))) (+) ( exp ( omega ,a)))] by A2;

        then P[(b (+) ((k *^ ( exp ( omega ,a))) (+) ( exp ( omega ,a))))] by Th81;

        then P[(b (+) ((k *^ ( exp ( omega ,a))) +^ ( exp ( omega ,a))))] by Lm9;

        then P[(b (+) (( succ k) *^ ( exp ( omega ,a))))] by ORDINAL2: 36;

        hence thesis by A16, Lm5;

      end;

      

       A17: for k be non zero Nat holds S[k] from NAT_1:sch 10( A12, A14);

      

       A18: for k be non zero Nat st R[k] holds R[(k + 1)]

      proof

        let k be non zero Nat;

        assume

         A19: R[k];

        let a be non empty Ordinal;

        assume

         A20: (k + 1) = ( len ( CantorNF a));

        consider c be Cantor-component Ordinal, A0 be Cantor-normal-form Ordinal-Sequence such that

         A21: ( CantorNF a) = ( <%c%> ^ A0) by ORDINAL5: 67;

        consider b be Ordinal, n be Nat such that

         A22: 0 in ( Segm n) & c = (n *^ ( exp ( omega ,b))) by ORDINAL5:def 9;

        reconsider n as non zero Nat by A22;

        

         A23: (k + 1) = (( len <%c%>) + ( len A0)) by A20, A21, AFINSQ_1: 17

        .= (1 + ( len A0)) by AFINSQ_1: 34;

        then A0 <> {} ;

        then

        reconsider a0 = ( Sum^ A0) as non empty Ordinal;

        ( len ( CantorNF a0)) = k by A23;

        then P[(a0 (+) (n *^ ( exp ( omega ,b))))] by A17, A19;

        then P[(( Sum^ <%c%>) (+) a0)] by A22, ORDINAL5: 53;

        then P[(( Sum^ <%c%>) +^ a0)] by A21, Th84;

        then P[(c +^ a0)] by ORDINAL5: 53;

        then P[( Sum^ ( <%c%> ^ A0))] by ORDINAL5: 55;

        hence thesis by A21;

      end;

      

       A24: for k be non zero Nat holds R[k] from NAT_1:sch 10( A8, A18);

      let a be non empty Ordinal;

      ( len ( CantorNF a)) is non zero;

      hence thesis by A24;

    end;

    theorem :: ORDINAL7:89

    

     Th102: for a,b be Ordinal st ( omega -exponent a) in ( omega -exponent b) holds a in ( exp ( omega ,( omega -exponent b)))

    proof

      defpred P[ non empty Ordinal] means for b be Ordinal st ( omega -exponent $1) in ( omega -exponent b) holds $1 in ( exp ( omega ,( omega -exponent b)));

      

       A1: for c be Ordinal, n be non zero Nat holds P[(n *^ ( exp ( omega ,c)))]

      proof

        let c be Ordinal, n be non zero Nat, b be Ordinal;

        assume

         A2: ( omega -exponent (n *^ ( exp ( omega ,c)))) in ( omega -exponent b);

         0 in n & n in omega by XBOOLE_1: 61, ORDINAL1: 11, ORDINAL1:def 12;

        then c in ( omega -exponent b) by A2, ORDINAL5: 58;

        then ( exp ( omega ,c)) in ( exp ( omega ,( omega -exponent b))) by ORDINAL4: 24;

        hence thesis by Th42;

      end;

      

       A3: for c be Ordinal, d be non empty Ordinal, n be non zero Nat st P[d] & not c in ( rng ( omega -exponent ( CantorNF d))) holds P[(d (+) (n *^ ( exp ( omega ,c))))]

      proof

        let c be Ordinal, d be non empty Ordinal, n be non zero Nat;

        assume

         A4: P[d] & not c in ( rng ( omega -exponent ( CantorNF d)));

        let b be Ordinal;

        assume ( omega -exponent (d (+) (n *^ ( exp ( omega ,c))))) in ( omega -exponent b);

        then (( omega -exponent d) \/ ( omega -exponent (n *^ ( exp ( omega ,c))))) in ( omega -exponent b) by Th100;

        then ( omega -exponent d) in ( omega -exponent b) & ( omega -exponent (n *^ ( exp ( omega ,c)))) in ( omega -exponent b) by XBOOLE_1: 7, ORDINAL1: 12;

        then d in ( exp ( omega ,( omega -exponent b))) & (n *^ ( exp ( omega ,c))) in ( exp ( omega ,( omega -exponent b))) by A1, A4;

        hence (d (+) (n *^ ( exp ( omega ,c)))) in ( exp ( omega ,( omega -exponent b))) by Th101;

      end;

      

       A5: for a be non empty Ordinal holds P[a] from OrdinalCNFIndA( A1, A3);

      let a,b be Ordinal;

      per cases ;

        suppose a <> {} ;

        hence thesis by A5;

      end;

        suppose

         A6: a = {} ;

        assume ( omega -exponent a) in ( omega -exponent b);

        thus thesis by A6, XBOOLE_1: 61, ORDINAL1: 11;

      end;

    end;

    theorem :: ORDINAL7:90

    

     Th103: for a,b be non empty Ordinal holds ( omega *^ a) c= b iff ( omega -exponent a) in ( omega -exponent b)

    proof

      let a,b be non empty Ordinal;

      

       A1: 0 in a & 0 in b & 1 in omega by XBOOLE_1: 61, ORDINAL1: 11;

      hereby

        assume

         A2: ( omega *^ a) c= b;

        ( exp ( omega ,( omega -exponent a))) c= a by A1, ORDINAL5:def 10;

        then ( omega *^ ( exp ( omega ,( omega -exponent a)))) c= ( omega *^ a) by ORDINAL2: 42;

        then ( exp ( omega ,( succ ( omega -exponent a)))) c= ( omega *^ a) by ORDINAL2: 44;

        then ( exp ( omega ,( succ ( omega -exponent a)))) c= b by A2, XBOOLE_1: 1;

        then ( succ ( omega -exponent a)) c= ( omega -exponent b) by A1, ORDINAL5:def 10;

        hence ( omega -exponent a) in ( omega -exponent b) by ORDINAL1: 6, TARSKI:def 3;

      end;

      assume ( omega -exponent a) in ( omega -exponent b);

      then

       A3: a in ( exp ( omega ,( omega -exponent b))) by Th102;

      reconsider fi = ( id omega ) as Ordinal-Sequence;

      

       A4: ( sup fi) = ( sup ( rng fi)) by ORDINAL2:def 5

      .= omega by ORDINAL2: 18;

      set psi = (fi *^ a);

      

       A5: ( dom fi) = ( dom psi) by ORDINAL3:def 4;

      for A,B be Ordinal st A in ( dom fi) & B = (fi . A) holds (psi . A) = (B *^ a) by ORDINAL3:def 4;

      then

       A6: ( sup psi) = ( omega *^ a) by A4, A5, ORDINAL3: 42;

      now

        let A be Ordinal;

        assume A in ( rng psi);

        then

        consider n be object such that

         A7: n in ( dom psi) & (psi . n) = A by FUNCT_1:def 3;

        reconsider n as Nat by A5, A7;

        A = ((fi . n) *^ a) by A5, A7, ORDINAL3:def 4

        .= (n *^ a) by A5, A7, FUNCT_1: 18;

        hence A in ( exp ( omega ,( omega -exponent b))) by A3, Th42;

      end;

      then ( sup ( rng psi)) c= ( exp ( omega ,( omega -exponent b))) by ORDINAL2: 20;

      then

       A8: ( omega *^ a) c= ( exp ( omega ,( omega -exponent b))) by A6, ORDINAL2:def 5;

       0 in b & 1 in omega by XBOOLE_1: 61, ORDINAL1: 11;

      then ( exp ( omega ,( omega -exponent b))) c= b by ORDINAL5:def 10;

      hence thesis by A8, XBOOLE_1: 1;

    end;

    theorem :: ORDINAL7:91

    for a,b be Ordinal st ( omega -exponent a) in ( omega -exponent b) holds (b -^ a) = b

    proof

      let a,b be Ordinal;

      assume

       A1: ( omega -exponent a) in ( omega -exponent b);

      per cases ;

        suppose a = 0 ;

        hence thesis by ORDINAL3: 56;

      end;

        suppose

         A2: a <> 0 ;

        

         A3: 1 in omega & 0 in b by A1, ORDINAL5:def 10;

        then ( omega *^ a) c= b by A1, A2, Th103;

        then

         A4: (a +^ b) = b by Th30;

        

         A5: a in ( exp ( omega ,( omega -exponent b))) by A1, Th102;

        ( exp ( omega ,( omega -exponent b))) c= b by A3, ORDINAL5:def 10;

        then a c= b by A5, ORDINAL1:def 2;

        hence thesis by A4, ORDINAL3:def 5;

      end;

    end;

    theorem :: ORDINAL7:92

    for a,b be Ordinal holds (a +^ b) c= (a (+) b)

    proof

      defpred P[ Nat] means for a,b be non empty Ordinal st ( len ( CantorNF a)) = $1 holds (a +^ b) c= (a (+) b);

      

       A1: P[1]

      proof

        let a,b be non empty Ordinal;

        assume ( len ( CantorNF a)) = 1;

        then

         A2: ( CantorNF a) = <%(( CantorNF a) . 0 )%> by AFINSQ_1: 34;

         0 in ( dom ( CantorNF a)) by XBOOLE_1: 61, ORDINAL1: 11;

        then (( CantorNF a) . 0 ) is Cantor-component by ORDINAL5:def 11;

        then

        consider c be Ordinal, m be Nat such that

         A3: 0 in ( Segm m) & (( CantorNF a) . 0 ) = (m *^ ( exp ( omega ,c))) by ORDINAL5:def 9;

        per cases by ORDINAL1: 16;

          suppose

           A4: ( omega -exponent b) c= c;

          (a +^ b) = (( Sum^ ( CantorNF a)) +^ b)

          .= ((( CantorNF a) . 0 ) +^ b) by A2, ORDINAL5: 53

          .= ((m *^ ( exp ( omega ,c))) (+) b) by A3, A4, Th83

          .= (( Sum^ ( CantorNF a)) (+) b) by A2, A3, ORDINAL5: 53

          .= (a (+) b);

          hence thesis;

        end;

          suppose

           A5: c in ( omega -exponent b);

           0 in m & m in omega by A3, ORDINAL1:def 12;

          then ( omega -exponent (( CantorNF a) . 0 )) in ( omega -exponent b) by A3, A5, ORDINAL5: 58;

          then ( omega -exponent ( Sum^ ( CantorNF a))) in ( omega -exponent b) by Th44;

          then ( omega *^ a) c= b by Th103;

          then (a +^ b) = b by Th30;

          hence (a +^ b) c= (a (+) b) by Th99;

        end;

      end;

      

       A6: for n be non zero Nat st P[n] holds P[(n + 1)]

      proof

        let n be non zero Nat;

        assume

         A7: P[n];

        let a,b be non empty Ordinal;

        assume

         A8: ( len ( CantorNF a)) = (n + 1);

        consider a0 be Cantor-component Ordinal, A0 be Cantor-normal-form Ordinal-Sequence such that

         A9: ( CantorNF a) = ( <%a0%> ^ A0) by ORDINAL5: 67;

        

         A10: (n + 1) = (( len <%a0%>) + ( len A0)) by A8, A9, AFINSQ_1: 17

        .= (( len ( CantorNF ( Sum^ A0))) + 1) by AFINSQ_1: 34;

        then ( CantorNF ( Sum^ A0)) <> {} ;

        then

         A11: (( Sum^ A0) +^ b) c= (( Sum^ A0) (+) b) by A7, A10;

        ( CantorNF a0) = <%a0%> by Th70;

        then ( len ( CantorNF a0)) = 1 by AFINSQ_1: 34;

        then

         A12: (a0 +^ (( Sum^ A0) (+) b)) c= (a0 (+) (( Sum^ A0) (+) b)) by A1;

        

         A13: a = ( Sum^ ( CantorNF a))

        .= (a0 +^ ( Sum^ A0)) by A9, ORDINAL5: 55

        .= (( Sum^ <%a0%>) +^ ( Sum^ A0)) by ORDINAL5: 53

        .= (( Sum^ <%a0%>) (+) ( Sum^ A0)) by A9, Th84

        .= (a0 (+) ( Sum^ A0)) by ORDINAL5: 53;

        (a +^ b) = (( Sum^ ( CantorNF a)) +^ b)

        .= ((a0 +^ ( Sum^ A0)) +^ b) by A9, ORDINAL5: 55

        .= (a0 +^ (( Sum^ A0) +^ b)) by ORDINAL3: 30;

        then (a +^ b) c= (a0 +^ (( Sum^ A0) (+) b)) by A11, ORDINAL2: 33;

        then (a +^ b) c= (a0 (+) (( Sum^ A0) (+) b)) by A12, XBOOLE_1: 1;

        hence (a +^ b) c= (a (+) b) by A13, Th81;

      end;

      

       A14: for n be non zero Nat holds P[n] from NAT_1:sch 10( A1, A6);

      let a,b be Ordinal;

      per cases ;

        suppose a = {} ;

        then (a +^ b) = b & (a (+) b) = b by Th82, ORDINAL2: 30;

        hence thesis;

      end;

        suppose b = {} ;

        then (a +^ b) = a & (a (+) b) = a by Th82, ORDINAL2: 27;

        hence thesis;

      end;

        suppose

         A15: a <> {} & b <> {} ;

        then ( len ( CantorNF a)) is non zero;

        hence thesis by A14, A15;

      end;

    end;

    theorem :: ORDINAL7:93

    for a,b,c be Ordinal st (a (+) b) = (a (+) c) holds b = c

    proof

      let a,b,c be Ordinal;

      assume

       A1: (a (+) b) = (a (+) c);

      set C1 = ( CantorNF (a (+) b)), C2 = ( CantorNF (a (+) c));

      set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

      set E3 = ( omega -exponent ( CantorNF c)), L1 = ( omega -leading_coeff ( CantorNF a));

      set L2 = ( omega -leading_coeff ( CantorNF b));

      set L3 = ( omega -leading_coeff ( CantorNF c));

      

       A2: ( rng E2) = ( rng E3)

      proof

        assume ( rng E2) <> ( rng E3);

        per cases by XBOOLE_0:def 10;

          suppose not ( rng E2) c= ( rng E3);

          then

          consider y be object such that

           A3: y in ( rng E2) & not y in ( rng E3) by TARSKI:def 3;

          y in (( rng E1) \/ ( rng E2)) by A3, XBOOLE_0:def 3;

          then

           A4: y in ( rng ( omega -exponent C1)) by Th76;

          then

          consider x be object such that

           A5: x in ( dom ( omega -exponent C1)) & (( omega -exponent C1) . x) = y by FUNCT_1:def 3;

          

           A6: x in ( dom C1) by A5, Def1;

          then

           A7: y = ( omega -exponent (C1 . x)) by A5, Def1;

          

           A8: ( omega -exponent (C1 . x)) in ( rng E1)

          proof

            assume not ( omega -exponent (C1 . x)) in ( rng E1);

            then not y in (( rng E1) \/ ( rng E3)) by A3, A7, XBOOLE_0:def 3;

            hence contradiction by A1, A4, Th76;

          end;

          then ( omega -exponent (C1 . x)) in (( rng E1) /\ ( rng E2)) by A3, A7, XBOOLE_0:def 4;

          then

           A9: ( omega -leading_coeff (C1 . x)) = ((L1 . ((E1 " ) . ( omega -exponent (C1 . x)))) + (L2 . ((E2 " ) . ( omega -exponent (C1 . x))))) by A6, Th80;

          ( omega -exponent (C1 . x)) in (( rng E1) \ ( rng E3)) by A3, A7, A8, XBOOLE_0:def 5;

          then ( omega -leading_coeff (C1 . x)) = ((L1 . ((E1 " ) . ( omega -exponent (C1 . x)))) + 0 ) by A1, A6, Th78;

          then

           A10: 0 = (L2 . ((E2 " ) . y)) by A7, A9;

          y in ( dom (E2 " )) by A3, FUNCT_1: 33;

          then ((E2 " ) . y) in ( rng (E2 " )) by FUNCT_1: 3;

          then ((E2 " ) . y) in ( dom E2) by FUNCT_1: 33;

          then ((E2 " ) . y) in ( dom ( CantorNF b)) by Def1;

          then ((E2 " ) . y) in ( dom L2) by Def3;

          hence contradiction by A10, FUNCT_1:def 9;

        end;

          suppose not ( rng E3) c= ( rng E2);

          then

          consider y be object such that

           A11: y in ( rng E3) & not y in ( rng E2) by TARSKI:def 3;

          y in (( rng E1) \/ ( rng E3)) by A11, XBOOLE_0:def 3;

          then

           A12: y in ( rng ( omega -exponent C2)) by Th76;

          then

          consider x be object such that

           A13: x in ( dom ( omega -exponent C2)) & (( omega -exponent C2) . x) = y by FUNCT_1:def 3;

          

           A14: x in ( dom C2) by A13, Def1;

          then

           A15: y = ( omega -exponent (C2 . x)) by A13, Def1;

          

           A16: ( omega -exponent (C2 . x)) in ( rng E1)

          proof

            assume not ( omega -exponent (C2 . x)) in ( rng E1);

            then not y in (( rng E1) \/ ( rng E2)) by A11, A15, XBOOLE_0:def 3;

            hence contradiction by A1, A12, Th76;

          end;

          then ( omega -exponent (C2 . x)) in (( rng E1) /\ ( rng E3)) by A11, A15, XBOOLE_0:def 4;

          then

           A17: ( omega -leading_coeff (C2 . x)) = ((L1 . ((E1 " ) . ( omega -exponent (C2 . x)))) + (L3 . ((E3 " ) . ( omega -exponent (C2 . x))))) by A14, Th80;

          ( omega -exponent (C2 . x)) in (( rng E1) \ ( rng E2)) by A11, A15, A16, XBOOLE_0:def 5;

          then ( omega -leading_coeff (C2 . x)) = ((L1 . ((E1 " ) . ( omega -exponent (C2 . x)))) + 0 ) by A1, A14, Th78;

          then

           A18: 0 = (L3 . ((E3 " ) . y)) by A15, A17;

          y in ( dom (E3 " )) by A11, FUNCT_1: 33;

          then ((E3 " ) . y) in ( rng (E3 " )) by FUNCT_1: 3;

          then ((E3 " ) . y) in ( dom E3) by FUNCT_1: 33;

          then ((E3 " ) . y) in ( dom ( CantorNF c)) by Def1;

          then ((E3 " ) . y) in ( dom L3) by Def3;

          hence contradiction by A18, FUNCT_1:def 9;

        end;

      end;

      then

       A19: E2 = E3 by Th34;

      

       A20: ( dom L2) = ( dom ( CantorNF b)) by Def3

      .= ( card ( dom E2)) by Def1

      .= ( card ( rng E2)) by CARD_1: 70

      .= ( card ( dom E3)) by A2, CARD_1: 70

      .= ( dom ( CantorNF c)) by Def1

      .= ( dom L3) by Def3;

      for x be object st x in ( dom L2) holds (L2 . x) = (L3 . x)

      proof

        let x be object;

        assume x in ( dom L2);

        then x in ( dom ( CantorNF b)) by Def3;

        then

         A21: x in ( dom E2) by Def1;

        then

         A22: (E2 . x) in ( rng E2) by FUNCT_1: 3;

        then (E2 . x) in (( rng E1) \/ ( rng E2)) by XBOOLE_0:def 3;

        then (E2 . x) in ( rng ( omega -exponent C1)) by Th76;

        then

        consider y be object such that

         A23: y in ( dom ( omega -exponent C1)) & (( omega -exponent C1) . y) = (E2 . x) by FUNCT_1:def 3;

        

         A24: y in ( dom C1) by A23, Def1;

        then

         A25: ( omega -exponent (C1 . y)) = (E2 . x) by A23, Def1;

        per cases ;

          suppose ( omega -exponent (C1 . y)) in ( rng E1);

          then

           A26: ( omega -exponent (C1 . y)) in (( rng E1) /\ ( rng E2)) by A22, A25, XBOOLE_0:def 4;

          then

           A27: ( omega -exponent (C2 . y)) in (( rng E1) /\ ( rng E3)) by A1, A2;

          ((L1 . ((E1 " ) . (E2 . x))) + (L2 . ((E2 " ) . (E2 . x)))) = ( omega -leading_coeff (C1 . y)) by A24, A25, A26, Th80

          .= ((L1 . ((E1 " ) . (E2 . x))) + (L3 . ((E3 " ) . (E2 . x)))) by A1, A24, A25, A27, Th80;

          

          hence (L2 . x) = (L3 . ((E3 " ) . (E2 . x))) by A21, FUNCT_1: 34

          .= (L3 . x) by A19, A21, FUNCT_1: 34;

        end;

          suppose not ( omega -exponent (C1 . y)) in ( rng E1);

          then

           A29: ( omega -exponent (C1 . y)) in (( rng E2) \ ( rng E1)) by A22, A25, XBOOLE_0:def 5;

          then

           A30: ( omega -exponent (C2 . y)) in (( rng E3) \ ( rng E1)) by A1, A2;

          

          thus (L2 . x) = (L2 . ((E2 " ) . (E2 . x))) by A21, FUNCT_1: 34

          .= ( omega -leading_coeff (C1 . y)) by A24, A25, A29, Th79

          .= (L3 . ((E3 " ) . (E2 . x))) by A1, A24, A25, A30, Th79

          .= (L3 . x) by A19, A21, FUNCT_1: 34;

        end;

      end;

      then L2 = L3 by A20, FUNCT_1: 2;

      then ( Sum^ ( CantorNF b)) = ( Sum^ ( CantorNF c)) by A19, Th67;

      hence thesis;

    end;

    

     Lm10: for A be Cantor-normal-form Ordinal-Sequence holds ( omega -exponent A) is XFinSequence of ( sup ( omega -exponent A))

    proof

      let A be Cantor-normal-form Ordinal-Sequence;

      now

        let y be object;

        assume y in ( rng ( omega -exponent A));

        then y in ( sup ( rng ( omega -exponent A))) by ORDINAL2: 19;

        hence y in ( sup ( omega -exponent A)) by ORDINAL2:def 5;

      end;

      hence thesis by TARSKI:def 3, RELAT_1:def 19;

    end;

    

     Lm11: for a be non empty Ordinal, b,c be Ordinal st b in c & (( CantorNF b) . 0 ) <> (( CantorNF c) . 0 ) holds (a (+) b) in (a (+) c)

    proof

      defpred P[ non empty Ordinal] means for a be non empty Ordinal, b be Ordinal st b in $1 holds (a (+) b) in (a (+) $1);

      let a be non empty Ordinal, b,d be Ordinal;

      assume

       A1: b in d & (( CantorNF b) . 0 ) <> (( CantorNF d) . 0 );

      then

       A2: ( omega -exponent b) c= ( omega -exponent d) by Th22, ORDINAL1:def 2;

      set c = ( omega -exponent d);

      defpred Q[ Nat] means $1 in ( dom ( CantorNF a)) & ( omega -exponent (( CantorNF a) . $1)) c= c & for j be Nat st j < $1 holds c in ( omega -exponent (( CantorNF a) . j));

      per cases ;

        suppose

         A3: for i be Nat holds not Q[i];

        defpred R[ Nat] means $1 in ( dom ( CantorNF a)) implies c in ( omega -exponent (( CantorNF a) . $1));

        

         A4: for k be Nat st for j be Nat st j < k holds R[j] holds R[k]

        proof

          let k be Nat;

          assume that

           A5: for j be Nat st j < k holds R[j] and

           A6: k in ( dom ( CantorNF a)) and

           A7: not c in ( omega -exponent (( CantorNF a) . k));

          

           A8: ( omega -exponent (( CantorNF a) . k)) c= c by A7, ORDINAL1: 16;

          for j be Nat st j < k holds c in ( omega -exponent (( CantorNF a) . j))

          proof

            let j be Nat;

            assume

             A9: j < k;

            then j in ( Segm k) by NAT_1: 44;

            then j in ( dom ( CantorNF a)) by A6, ORDINAL1: 10;

            hence thesis by A5, A9;

          end;

          hence contradiction by A3, A6, A8;

        end;

        

         A10: for k be Nat holds R[k] from NAT_1:sch 4( A4);

        consider A0 be Cantor-normal-form Ordinal-Sequence, a0 be Cantor-component Ordinal such that

         A11: ( CantorNF a) = (A0 ^ <%a0%>) by Th29;

        ( len ( CantorNF a)) = (( len A0) + ( len <%a0%>)) by A11, AFINSQ_1: 17

        .= (( len A0) + 1) by AFINSQ_1: 34;

        then ( len A0) < ( len ( CantorNF a)) by NAT_1: 13;

        then ( len A0) in ( Segm ( len ( CantorNF a))) by NAT_1: 44;

        then c in ( omega -exponent (( CantorNF a) . ( len A0))) by A10;

        then c in ( omega -exponent a0) by A11, AFINSQ_1: 36;

        then

         A12: c in ( omega -exponent ( last ( CantorNF a))) by A11, AFINSQ_1: 92;

        then

         A13: (a (+) d) = (a +^ d) by Th85;

        (a (+) b) = (a +^ b) by A2, A12, Th85, ORDINAL1: 12;

        hence thesis by A1, A13, ORDINAL2: 32;

      end;

        suppose ex i be Nat st Q[i];

        then

        consider i be Nat such that

         A14: Q[i];

        set C1 = ( CantorNF (a (+) b)), C2 = ( CantorNF (a (+) d));

        set A1 = (C1 | i), A2 = (C2 | i), B1 = (C1 /^ i), B2 = (C2 /^ i);

        set E1 = ( omega -exponent ( CantorNF a)), E2 = ( omega -exponent ( CantorNF b));

        set E3 = ( omega -exponent ( CantorNF d));

        set L1 = ( omega -leading_coeff ( CantorNF a));

        set L2 = ( omega -leading_coeff ( CantorNF b));

        set L3 = ( omega -leading_coeff ( CantorNF d));

        

         A15: 0 in ( dom ( CantorNF d)) by A1, XBOOLE_1: 61, ORDINAL1: 11;

        then

         A16: 0 in ( dom E3) by Def1;

        then (E3 . 0 ) in ( rng E3) by FUNCT_1: 3;

        then ( omega -exponent (( CantorNF d) . 0 )) in ( rng E3) by A15, Def1;

        then

         A17: ( omega -exponent ( Sum^ ( CantorNF d))) in ( rng E3) by Th44;

        then

         A18: c in (( rng E1) \/ ( rng E3)) by XBOOLE_0:def 3;

        then

         A19: c in ( rng ( omega -exponent C2)) by Th76;

        

         A20: c = ( omega -exponent ( Sum^ ( CantorNF d)))

        .= ( omega -exponent (( CantorNF d) . 0 )) by Th44

        .= (E3 . 0 ) by A15, Def1;

        

         A21: i in ( dom E1) by A14, Def1;

        

         A22: i in ( dom C1) & i in ( dom C2) by A14, Th77, TARSKI:def 3;

        ( omega -exponent C2) = ( omega -exponent (A2 ^ B2))

        .= (( omega -exponent A2) ^ ( omega -exponent B2)) by Th47

        .= ((( omega -exponent C2) | i) ^ ( omega -exponent B2)) by Th48

        .= ((( omega -exponent C2) | i) ^ (( omega -exponent C2) /^ i)) by Th49;

        then

         A23: ( rng ( omega -exponent C2)) = (( rng (( omega -exponent C2) | i)) \/ ( rng (( omega -exponent C2) /^ i))) by Th9;

        ( omega -exponent C2) is XFinSequence of ( sup ( omega -exponent C2)) by Lm10;

        then

         A24: ( rng (( omega -exponent C2) | i)) misses ( rng (( omega -exponent C2) /^ i)) by Th19;

        

         A25: ( rng ( omega -exponent B2)) = ( rng (( omega -exponent C2) /^ i)) by Th49

        .= (( rng ( omega -exponent C2)) \ ( rng (( omega -exponent C2) | i))) by A23, A24, XBOOLE_1: 88

        .= ((( rng E1) \/ ( rng E3)) \ ( rng (( omega -exponent C2) | i))) by Th76

        .= ((( rng E1) \/ ( rng E3)) \ ( rng ( omega -exponent A2))) by Th48;

        ( omega -exponent C1) = ( omega -exponent (A1 ^ B1))

        .= (( omega -exponent A1) ^ ( omega -exponent B1)) by Th47

        .= ((( omega -exponent C1) | i) ^ ( omega -exponent B1)) by Th48

        .= ((( omega -exponent C1) | i) ^ (( omega -exponent C1) /^ i)) by Th49;

        then

         A26: ( rng ( omega -exponent C1)) = (( rng (( omega -exponent C1) | i)) \/ ( rng (( omega -exponent C1) /^ i))) by Th9;

        ( omega -exponent C1) is XFinSequence of ( sup ( omega -exponent C1)) by Lm10;

        then

         A27: ( rng (( omega -exponent C1) | i)) misses ( rng (( omega -exponent C1) /^ i)) by Th19;

        

         A28: ( rng ( omega -exponent B1)) = ( rng (( omega -exponent C1) /^ i)) by Th49

        .= (( rng ( omega -exponent C1)) \ ( rng (( omega -exponent C1) | i))) by A26, A27, XBOOLE_1: 88

        .= ((( rng E1) \/ ( rng E2)) \ ( rng (( omega -exponent C1) | i))) by Th76

        .= ((( rng E1) \/ ( rng E2)) \ ( rng ( omega -exponent A1))) by Th48;

        

         A29: ( dom A1) = ( dom A2)

        proof

          

           A30: i in ( dom C1) & i in ( dom C2) by A14, Th77, TARSKI:def 3;

          now

            let x be object;

            hereby

              assume x in ( dom A1);

              then

               A31: x in i by RELAT_1: 57;

              then x in ( dom C2) by A30, ORDINAL1: 10;

              hence x in ( dom A2) by A31, RELAT_1: 57;

            end;

            assume x in ( dom A2);

            then

             A32: x in i by RELAT_1: 57;

            then x in ( dom C1) by A30, ORDINAL1: 10;

            hence x in ( dom A1) by A32, RELAT_1: 57;

          end;

          hence thesis by TARSKI: 2;

        end;

        

         A33: for n be Nat st n in ( dom A1) holds (A1 . n) = (( CantorNF a) . n)

        proof

          defpred R[ Nat] means $1 in ( dom A1) & (A1 . $1) <> (( CantorNF a) . $1);

          assume

           A34: ex n be Nat st R[n];

          consider n be Nat such that

           A35: R[n] & for m be Nat st R[m] holds n <= m from NAT_1:sch 5( A34);

          

           A36: n in i by A35, RELAT_1: 57;

          then

           A37: n in ( dom ( CantorNF a)) by A14, ORDINAL1: 10;

          then

           A38: n in ( dom C1) by Th77, TARSKI:def 3;

          

           A39: not ( omega -exponent (C1 . n)) in ( rng E2)

          proof

            assume ( omega -exponent (C1 . n)) in ( rng E2);

            then (( omega -exponent C1) . n) in ( rng E2) by A38, Def1;

            then

            consider m be object such that

             A40: m in ( dom E2) & (E2 . m) = (( omega -exponent C1) . n) by FUNCT_1:def 3;

            reconsider m as Nat by A40;

            n in ( Segm i) by A36;

            then c in ( omega -exponent (( CantorNF a) . n)) by A14, NAT_1: 44;

            then c in (E1 . n) by A37, Def1;

            then ( omega -exponent ( Sum^ ( CantorNF b))) in (E1 . n) by A2, ORDINAL1: 12;

            then ( omega -exponent (( CantorNF b) . 0 )) in (E1 . n) by Th44;

            then

             A42: ( omega -exponent (( CantorNF b) . 0 )) in (E2 . m) by A40, Th97, TARSKI:def 3;

            

             A43: m in ( dom ( CantorNF b)) by A40, Def1;

            then

             A44: ( omega -exponent (( CantorNF b) . 0 )) in ( omega -exponent (( CantorNF b) . m)) by A42, Def1;

            then not 0 in m by A43, ORDINAL5:def 11;

            then m = 0 by ORDINAL1: 16, XBOOLE_1: 3;

            hence contradiction by A44;

          end;

          

           A45: ( omega -exponent (C1 . n)) = (E1 . n)

          proof

            assume ( omega -exponent (C1 . n)) <> (E1 . n);

            then (( omega -exponent C1) . n) <> (E1 . n) by A38, Def1;

            then (E1 . n) c< (( omega -exponent C1) . n) by Th97, XBOOLE_0:def 8;

            then

             A46: (E1 . n) in (( omega -exponent C1) . n) by ORDINAL1: 11;

            n in ( dom ( omega -exponent C1)) by A38, Def1;

            then (( omega -exponent C1) . n) in ( rng ( omega -exponent C1)) by FUNCT_1: 3;

            then

             A47: (( omega -exponent C1) . n) in (( rng E1) \/ ( rng E2)) by Th76;

             not (( omega -exponent C1) . n) in ( rng E2) by A38, A39, Def1;

            then (( omega -exponent C1) . n) in ( rng E1) by A47, XBOOLE_0:def 3;

            then

            consider m be object such that

             A48: m in ( dom E1) & (E1 . m) = (( omega -exponent C1) . n) by FUNCT_1:def 3;

            reconsider m as Nat by A48;

            

             A49: m in n

            proof

              assume not m in n;

              per cases by ORDINAL1: 14;

                suppose m = n;

                hence contradiction by A46, A48;

              end;

                suppose

                 A50: n in m;

                

                 A51: m in ( dom ( CantorNF a)) by A48, Def1;

                then ( omega -exponent (( CantorNF a) . m)) in ( omega -exponent (( CantorNF a) . n)) by A50, ORDINAL5:def 11;

                then (E1 . m) in ( omega -exponent (( CantorNF a) . n)) by A51, Def1;

                hence contradiction by A37, A46, A48, Def1;

              end;

            end;

            then

             A52: m in ( Segm n);

            

             A53: m in ( dom A1) by A35, A49, ORDINAL1: 10;

            

             A54: m in ( dom ( CantorNF a)) by A37, A49, ORDINAL1: 10;

            

             A55: m in ( dom C1) by A38, A49, ORDINAL1: 10;

            

             A56: (( omega -exponent C1) . m) = ( omega -exponent (C1 . m)) by A38, A49, Def1, ORDINAL1: 10

            .= ( omega -exponent (A1 . m)) by A53, FUNCT_1: 47

            .= ( omega -exponent (( CantorNF a) . m)) by A35, A52, A53, NAT_1: 44

            .= (E1 . m) by A54, Def1;

            ( omega -exponent (C1 . n)) in ( omega -exponent (C1 . m)) by A38, A49, ORDINAL5:def 11;

            then (( omega -exponent C1) . n) in ( omega -exponent (C1 . m)) by A38, Def1;

            then (( omega -exponent C1) . n) in (( omega -exponent C1) . m) by A55, Def1;

            hence contradiction by A48, A56;

          end;

          

           A57: n in ( dom E1) by A37, Def1;

          n in ( dom E1) by A37, Def1;

          then ( omega -exponent (C1 . n)) in ( rng E1) by A45, FUNCT_1: 3;

          then ( omega -exponent (C1 . n)) in (( rng E1) \ ( rng E2)) by A39, XBOOLE_0:def 5;

          

          then

           A58: ( omega -leading_coeff (C1 . n)) = (L1 . ((E1 " ) . (E1 . n))) by A38, A45, Th78

          .= (L1 . n) by A57, FUNCT_1: 34;

          (A1 . n) = (C1 . n) by A36, FUNCT_1: 49

          .= ((L1 . n) *^ ( exp ( omega ,( omega -exponent (C1 . n))))) by A38, A58, Th64

          .= (( CantorNF a) . n) by A37, A45, Th65;

          hence contradiction by A35;

        end;

        

         A59: for n be Nat st n in ( dom A2) holds (A2 . n) = (( CantorNF a) . n)

        proof

          defpred R[ Nat] means $1 in ( dom A2) & (A2 . $1) <> (( CantorNF a) . $1);

          assume

           A60: ex n be Nat st R[n];

          consider n be Nat such that

           A61: R[n] & for m be Nat st R[m] holds n <= m from NAT_1:sch 5( A60);

          

           A62: n in i by A61, RELAT_1: 57;

          then

           A63: n in ( dom ( CantorNF a)) by A14, ORDINAL1: 10;

          then

           A64: n in ( dom C2) by Th77, TARSKI:def 3;

          

           A65: not ( omega -exponent (C2 . n)) in ( rng E3)

          proof

            assume ( omega -exponent (C2 . n)) in ( rng E3);

            then (( omega -exponent C2) . n) in ( rng E3) by A64, Def1;

            then

            consider m be object such that

             A66: m in ( dom E3) & (E3 . m) = (( omega -exponent C2) . n) by FUNCT_1:def 3;

            reconsider m as Nat by A66;

            n in ( Segm i) by A62;

            then c in ( omega -exponent (( CantorNF a) . n)) by A14, NAT_1: 44;

            then ( omega -exponent ( Sum^ ( CantorNF d))) in (E1 . n) by A63, Def1;

            then ( omega -exponent (( CantorNF d) . 0 )) in (E1 . n) by Th44;

            then

             A67: ( omega -exponent (( CantorNF d) . 0 )) in (E3 . m) by A66, Th97, TARSKI:def 3;

            

             A68: m in ( dom ( CantorNF d)) by A66, Def1;

            then

             A69: ( omega -exponent (( CantorNF d) . 0 )) in ( omega -exponent (( CantorNF d) . m)) by A67, Def1;

            then not 0 in m by A68, ORDINAL5:def 11;

            then m = 0 by ORDINAL1: 16, XBOOLE_1: 3;

            hence contradiction by A69;

          end;

          

           A70: ( omega -exponent (C2 . n)) = (E1 . n)

          proof

            assume ( omega -exponent (C2 . n)) <> (E1 . n);

            then (( omega -exponent C2) . n) <> (E1 . n) by A64, Def1;

            then (E1 . n) c< (( omega -exponent C2) . n) by Th97, XBOOLE_0:def 8;

            then

             A71: (E1 . n) in (( omega -exponent C2) . n) by ORDINAL1: 11;

            n in ( dom ( omega -exponent C2)) by A64, Def1;

            then (( omega -exponent C2) . n) in ( rng ( omega -exponent C2)) by FUNCT_1: 3;

            then

             A72: (( omega -exponent C2) . n) in (( rng E1) \/ ( rng E3)) by Th76;

             not (( omega -exponent C2) . n) in ( rng E3) by A64, A65, Def1;

            then (( omega -exponent C2) . n) in ( rng E1) by A72, XBOOLE_0:def 3;

            then

            consider m be object such that

             A73: m in ( dom E1) & (E1 . m) = (( omega -exponent C2) . n) by FUNCT_1:def 3;

            reconsider m as Nat by A73;

            

             A74: m in n

            proof

              assume not m in n;

              per cases by ORDINAL1: 14;

                suppose m = n;

                hence contradiction by A71, A73;

              end;

                suppose

                 A75: n in m;

                

                 A76: m in ( dom ( CantorNF a)) by A73, Def1;

                then ( omega -exponent (( CantorNF a) . m)) in ( omega -exponent (( CantorNF a) . n)) by A75, ORDINAL5:def 11;

                then (E1 . m) in ( omega -exponent (( CantorNF a) . n)) by A76, Def1;

                hence contradiction by A63, A71, A73, Def1;

              end;

            end;

            then m in ( Segm n);

            then

             A77: m < n by NAT_1: 44;

            

             A78: m in ( dom A2) by A61, A74, ORDINAL1: 10;

            

             A79: m in ( dom ( CantorNF a)) by A63, A74, ORDINAL1: 10;

            

             A80: m in ( dom C2) by A64, A74, ORDINAL1: 10;

            

             A81: (( omega -exponent C2) . m) = ( omega -exponent (C2 . m)) by A64, A74, Def1, ORDINAL1: 10

            .= ( omega -exponent (A2 . m)) by A78, FUNCT_1: 47

            .= ( omega -exponent (( CantorNF a) . m)) by A61, A77, A78

            .= (E1 . m) by A79, Def1;

            ( omega -exponent (C2 . n)) in ( omega -exponent (C2 . m)) by A64, A74, ORDINAL5:def 11;

            then (( omega -exponent C2) . n) in ( omega -exponent (C2 . m)) by A64, Def1;

            then (( omega -exponent C2) . n) in (( omega -exponent C2) . m) by A80, Def1;

            hence contradiction by A73, A81;

          end;

          

           A82: n in ( dom E1) by A63, Def1;

          n in ( dom E1) by A63, Def1;

          then ( omega -exponent (C2 . n)) in ( rng E1) by A70, FUNCT_1: 3;

          then ( omega -exponent (C2 . n)) in (( rng E1) \ ( rng E3)) by A65, XBOOLE_0:def 5;

          

          then

           A83: ( omega -leading_coeff (C2 . n)) = (L1 . ((E1 " ) . ( omega -exponent (C2 . n)))) by A64, Th78

          .= (L1 . n) by A70, A82, FUNCT_1: 34;

          (A2 . n) = (C2 . n) by A62, FUNCT_1: 49

          .= ((L1 . n) *^ ( exp ( omega ,( omega -exponent (C2 . n))))) by A64, A83, Th64

          .= (( CantorNF a) . n) by A63, A70, Th65;

          hence contradiction by A61;

        end;

        for x be object st x in ( dom A1) holds (A1 . x) = (A2 . x)

        proof

          let x be object;

          assume x in ( dom A1);

          then (A1 . x) = (( CantorNF a) . x) & (A2 . x) = (( CantorNF a) . x) by A29, A33, A59;

          hence thesis;

        end;

        then

         A85: ( Sum^ A1) = ( Sum^ A2) by A29, FUNCT_1: 2;

        

         A86: ( omega -exponent (C2 . i)) = c

        proof

          assume

           A87: ( omega -exponent (C2 . i)) <> c;

          

           A88: not ( omega -exponent (C2 . i)) in ( rng E3)

          proof

             not ( omega -exponent (C2 . i)) in c

            proof

              assume

               A89: ( omega -exponent (C2 . i)) in c;

              consider j be object such that

               A90: j in ( dom ( omega -exponent C2)) & (( omega -exponent C2) . j) = c by A19, FUNCT_1:def 3;

              reconsider j as Nat by A90;

              

               A91: j in ( dom C2) by A90, Def1;

              then

               A92: ( omega -exponent (C2 . j)) = c by A90, Def1;

              per cases by ORDINAL1: 14;

                suppose

                 A93: j in i;

                then

                 A94: j in ( Segm i);

                (( CantorNF a) . j) = (A2 . j) by A59, A91, A93, RELAT_1: 57

                .= (C2 . j) by A93, FUNCT_1: 49;

                then

                 A95: c = ( omega -exponent (( CantorNF a) . j)) by A92;

                c in ( omega -exponent (( CantorNF a) . j)) by A14, A94, NAT_1: 44;

                hence contradiction by A95;

              end;

                suppose j = i;

                hence contradiction by A89, A92;

              end;

                suppose i in j;

                hence contradiction by A89, A91, A92, ORDINAL5:def 11;

              end;

            end;

            then

             A96: c in ( omega -exponent (C2 . i)) by A87, ORDINAL1: 14;

            assume ( omega -exponent (C2 . i)) in ( rng E3);

            then

            consider k be object such that

             A97: k in ( dom E3) & (E3 . k) = ( omega -exponent (C2 . i)) by FUNCT_1:def 3;

            reconsider k as Nat by A97;

            ( omega -exponent ( Sum^ ( CantorNF d))) in (E3 . k) by A96, A97;

            then ( omega -exponent (( CantorNF d) . 0 )) in (E3 . k) by Th44;

            then

             A98: (E3 . 0 ) in (E3 . k) by A15, Def1;

            per cases ;

              suppose k = 0 ;

              hence contradiction by A98;

            end;

              suppose 0 < k;

              then 0 in ( Segm k) by NAT_1: 44;

              hence contradiction by A97, A98, ORDINAL5:def 1;

            end;

          end;

          ( omega -exponent (C2 . i)) = (E1 . i)

          proof

            assume

             A100: ( omega -exponent (C2 . i)) <> (E1 . i);

            i in ( dom ( omega -exponent C2)) by A22, Def1;

            then (( omega -exponent C2) . i) in ( rng ( omega -exponent C2)) by FUNCT_1: 3;

            then ( omega -exponent (C2 . i)) in ( rng ( omega -exponent C2)) by A22, Def1;

            then ( omega -exponent (C2 . i)) in (( rng E1) \/ ( rng E3)) by Th76;

            then ( omega -exponent (C2 . i)) in ( rng E1) by A88, XBOOLE_0:def 3;

            then

            consider j be object such that

             A101: j in ( dom E1) & (E1 . j) = ( omega -exponent (C2 . i)) by FUNCT_1:def 3;

            reconsider j as Nat by A101;

            per cases by XXREAL_0: 1;

              suppose j < i;

              then

               A102: j in ( Segm i) by NAT_1: 44;

              then

               A103: j in ( dom ( CantorNF a)) by A14, ORDINAL1: 10;

              i in ( dom C1) & i in ( dom C2) by A14, Th77, TARSKI:def 3;

              then j in ( dom C1) & j in ( dom C2) by A102, ORDINAL1: 10;

              then

               A104: j in ( dom A1) & j in ( dom A2) by A102, RELAT_1: 57;

              

              then

               A105: ( omega -exponent (C2 . j)) = ( omega -exponent (A2 . j)) by FUNCT_1: 47

              .= ( omega -exponent (( CantorNF a) . j)) by A59, A104

              .= ( omega -exponent (C2 . i)) by A101, A103, Def1;

              i in ( dom C2) by A14, Th77, TARSKI:def 3;

              then ( omega -exponent (C2 . i)) in ( omega -exponent (C2 . j)) by A102, ORDINAL5:def 11;

              hence contradiction by A105;

            end;

              suppose j = i;

              hence contradiction by A100, A101;

            end;

              suppose j > i;

              then i in ( Segm j) by NAT_1: 44;

              then

               A106: i in j;

              

               A107: j in ( dom ( CantorNF a)) by A101, Def1;

              then ( omega -exponent (( CantorNF a) . j)) in ( omega -exponent (( CantorNF a) . i)) by A106, ORDINAL5:def 11;

              then (E1 . j) in ( omega -exponent (( CantorNF a) . i)) by A107, Def1;

              then

               A108: ( omega -exponent (C2 . i)) in (E1 . i) by A14, A101, Def1;

              (E1 . i) in ( rng E1) by A21, FUNCT_1: 3;

              then

               A109: (E1 . i) in (( rng E1) \/ ( rng E3)) by XBOOLE_1: 7, TARSKI:def 3;

               not (E1 . i) in ( rng ( omega -exponent A2))

              proof

                assume (E1 . i) in ( rng ( omega -exponent A2));

                then

                consider k be object such that

                 A110: k in ( dom ( omega -exponent A2)) & (( omega -exponent A2) . k) = (E1 . i) by FUNCT_1:def 3;

                k in ( dom A2) by A110, Def1;

                then

                 A111: k in i & k in ( dom C2) by RELAT_1: 57;

                

                 A112: (E1 . i) = ((( omega -exponent C2) | i) . k) by A110, Th48

                .= (( omega -exponent C2) . k) by A111, FUNCT_1: 49

                .= ( omega -exponent (C2 . k)) by A111, Def1;

                

                 A113: ( omega -exponent (C2 . i)) in ( omega -exponent (C2 . k)) by A22, A111, ORDINAL5:def 11;

                (E1 . i) c= (( omega -exponent C2) . i) by Th97;

                then (E1 . i) c= ( omega -exponent (C2 . i)) by A22, Def1;

                hence contradiction by A112, A113, ORDINAL1: 12;

              end;

              then (E1 . i) in ( rng ( omega -exponent B2)) by A25, A109, XBOOLE_0:def 5;

              then

              consider k be object such that

               A114: k in ( dom ( omega -exponent B2)) & (( omega -exponent B2) . k) = (E1 . i) by FUNCT_1:def 3;

              reconsider k as Nat by A114;

              

               A115: k in ( dom B2) by A114, Def1;

              

              then

               A116: (E1 . i) = ( omega -exponent (B2 . k)) by A114, Def1

              .= ( omega -exponent (C2 . (k + i))) by A115, AFINSQ_2:def 2;

              per cases ;

                suppose k = 0 ;

                hence contradiction by A108, A116;

              end;

                suppose 0 < k;

                then ( 0 + i) < (k + i) by XREAL_1: 8;

                then i in ( Segm (k + i)) by NAT_1: 44;

                then

                 A117: i in (k + i);

                k in ( Segm ( len B2)) by A115;

                then k < ( len B2) by NAT_1: 44;

                then k < (( len C2) -' i) by AFINSQ_2:def 2;

                then

                 A118: (k + i) < ((( len C2) -' i) + i) by XREAL_1: 8;

                i in ( Segm ( len C2)) by A22;

                then i < ( len C2) by NAT_1: 44;

                then (k + i) < ( len C2) by A118, XREAL_1: 235;

                then (k + i) in ( Segm ( len C2)) by NAT_1: 44;

                hence contradiction by A108, A116, A117, ORDINAL5:def 11;

              end;

            end;

          end;

          then ( omega -exponent (C2 . i)) c= c by A14, Def1;

          then

           A119: ( omega -exponent (C2 . i)) in c by A87, XBOOLE_0:def 8, ORDINAL1: 11;

           not c in ( rng ( omega -exponent A2))

          proof

            assume c in ( rng ( omega -exponent A2));

            then

            consider m be object such that

             A121: m in ( dom ( omega -exponent A2)) & (( omega -exponent A2) . m) = c by FUNCT_1:def 3;

            reconsider m as Nat by A121;

            

             A122: m in ( dom A2) by A121, Def1;

            then

             A123: m in ( dom C2) & m in i by RELAT_1: 57;

            c = ( omega -exponent (A2 . m)) by A121, A122, Def1

            .= ( omega -exponent (C2 . m)) by A122, FUNCT_1: 47

            .= (( omega -exponent C2) . m) by A123, Def1;

            then (E1 . m) c= c by Th97;

            then

             A125: ( omega -exponent (( CantorNF a) . m)) c= c by A14, A123, Def1, ORDINAL1: 10;

            m in ( Segm i) by A123;

            then c in ( omega -exponent (( CantorNF a) . m)) by A14, NAT_1: 44;

            then c in c by A125;

            hence contradiction;

          end;

          then c in ( rng ( omega -exponent B2)) by A18, A25, XBOOLE_0:def 5;

          then

          consider m be object such that

           A126: m in ( dom ( omega -exponent B2)) & (( omega -exponent B2) . m) = c by FUNCT_1:def 3;

          reconsider m as Nat by A126;

          

           A127: m in ( dom B2) by A126, Def1;

          

          then

           A128: c = ( omega -exponent (B2 . m)) by A126, Def1

          .= ( omega -exponent (C2 . (m + i))) by A127, AFINSQ_2:def 2;

          per cases ;

            suppose m = 0 ;

            hence contradiction by A119, A128;

          end;

            suppose m > 0 ;

            then ( 0 + i) < (m + i) by XREAL_1: 8;

            then i in ( Segm (m + i)) by NAT_1: 44;

            then

             A129: i in (m + i);

            m in ( Segm ( len B2)) by A127;

            then m < ( len B2) by NAT_1: 44;

            then m < (( len C2) -' i) by AFINSQ_2:def 2;

            then

             A130: (m + i) < ((( len C2) -' i) + i) by XREAL_1: 8;

            i in ( Segm ( len C2)) by A22;

            then i < ( len C2) by NAT_1: 44;

            then (m + i) < ( len C2) by A130, XREAL_1: 235;

            then (m + i) in ( Segm ( len C2)) by NAT_1: 44;

            hence contradiction by A119, A128, A129, ORDINAL5:def 11;

          end;

        end;

        

         A131: ( omega -exponent b) = c implies ( omega -exponent (C1 . i)) = c

        proof

          assume

           A132: ( omega -exponent b) = c;

          per cases ;

            suppose b <> {} ;

            then

             A133: 0 in ( dom ( CantorNF b)) by XBOOLE_1: 61, ORDINAL1: 11;

            then 0 in ( dom E2) by Def1;

            then (E2 . 0 ) in ( rng E2) by FUNCT_1: 3;

            then ( omega -exponent (( CantorNF b) . 0 )) in ( rng E2) by A133, Def1;

            then ( omega -exponent ( Sum^ ( CantorNF b))) in ( rng E2) by Th44;

            then

             A134: c in (( rng E1) \/ ( rng E2)) by A132, XBOOLE_0:def 3;

            then

             A135: c in ( rng ( omega -exponent C1)) by Th76;

            assume

             A136: ( omega -exponent (C1 . i)) <> c;

            

             A137: not ( omega -exponent (C1 . i)) in ( rng E2)

            proof

               not ( omega -exponent (C1 . i)) in c

              proof

                assume

                 A138: ( omega -exponent (C1 . i)) in c;

                consider j be object such that

                 A139: j in ( dom ( omega -exponent C1)) & (( omega -exponent C1) . j) = c by A135, FUNCT_1:def 3;

                reconsider j as Nat by A139;

                

                 A140: j in ( dom C1) by A139, Def1;

                then

                 A141: ( omega -exponent (C1 . j)) = c by A139, Def1;

                per cases by ORDINAL1: 14;

                  suppose

                   A142: j in i;

                  then

                   A143: j in ( Segm i);

                  (( CantorNF a) . j) = (A1 . j) by A33, A140, A142, RELAT_1: 57

                  .= (C1 . j) by A142, FUNCT_1: 49;

                  then

                   A144: c = ( omega -exponent (( CantorNF a) . j)) by A141;

                  c in ( omega -exponent (( CantorNF a) . j)) by A14, A143, NAT_1: 44;

                  hence contradiction by A144;

                end;

                  suppose j = i;

                  hence contradiction by A138, A141;

                end;

                  suppose i in j;

                  hence contradiction by A138, A140, A141, ORDINAL5:def 11;

                end;

              end;

              then

               A145: c in ( omega -exponent (C1 . i)) by A136, ORDINAL1: 14;

              assume ( omega -exponent (C1 . i)) in ( rng E2);

              then

              consider k be object such that

               A146: k in ( dom E2) & (E2 . k) = ( omega -exponent (C1 . i)) by FUNCT_1:def 3;

              reconsider k as Nat by A146;

               0 in ( dom E2) by A146, XBOOLE_1: 61, ORDINAL1: 11;

              then

               A147: 0 in ( dom ( CantorNF b)) by Def1;

              ( omega -exponent ( Sum^ ( CantorNF b))) in (E2 . k) by A132, A145, A146;

              then ( omega -exponent (( CantorNF b) . 0 )) in (E2 . k) by Th44;

              then

               A148: (E2 . 0 ) in (E2 . k) by A147, Def1;

              per cases ;

                suppose k = 0 ;

                hence contradiction by A148;

              end;

                suppose 0 < k;

                then 0 in ( Segm k) by NAT_1: 44;

                hence contradiction by A146, A148, ORDINAL5:def 1;

              end;

            end;

            ( omega -exponent (C1 . i)) = (E1 . i)

            proof

              assume

               A150: ( omega -exponent (C1 . i)) <> (E1 . i);

              i in ( dom ( omega -exponent C1)) by A22, Def1;

              then (( omega -exponent C1) . i) in ( rng ( omega -exponent C1)) by FUNCT_1: 3;

              then ( omega -exponent (C1 . i)) in ( rng ( omega -exponent C1)) by A22, Def1;

              then ( omega -exponent (C1 . i)) in (( rng E1) \/ ( rng E2)) by Th76;

              then ( omega -exponent (C1 . i)) in ( rng E1) by A137, XBOOLE_0:def 3;

              then

              consider j be object such that

               A151: j in ( dom E1) & (E1 . j) = ( omega -exponent (C1 . i)) by FUNCT_1:def 3;

              reconsider j as Nat by A151;

              per cases by XXREAL_0: 1;

                suppose j < i;

                then

                 A152: j in ( Segm i) by NAT_1: 44;

                then

                 A153: j in ( dom ( CantorNF a)) by A14, ORDINAL1: 10;

                i in ( dom C2) & i in ( dom C1) by A14, Th77, TARSKI:def 3;

                then j in ( dom C2) & j in ( dom C1) by A152, ORDINAL1: 10;

                then

                 A154: j in ( dom A1) & j in ( dom A2) by A152, RELAT_1: 57;

                

                then

                 A155: ( omega -exponent (C1 . j)) = ( omega -exponent (A1 . j)) by FUNCT_1: 47

                .= ( omega -exponent (( CantorNF a) . j)) by A33, A154

                .= ( omega -exponent (C1 . i)) by A151, A153, Def1;

                i in ( dom C1) by A14, Th77, TARSKI:def 3;

                then ( omega -exponent (C1 . i)) in ( omega -exponent (C1 . j)) by A152, ORDINAL5:def 11;

                hence contradiction by A155;

              end;

                suppose j = i;

                hence contradiction by A150, A151;

              end;

                suppose j > i;

                then i in ( Segm j) by NAT_1: 44;

                then

                 A156: i in j;

                

                 A157: j in ( dom ( CantorNF a)) by A151, Def1;

                then ( omega -exponent (( CantorNF a) . j)) in ( omega -exponent (( CantorNF a) . i)) by A156, ORDINAL5:def 11;

                then (E1 . j) in ( omega -exponent (( CantorNF a) . i)) by A157, Def1;

                then

                 A158: ( omega -exponent (C1 . i)) in (E1 . i) by A14, A151, Def1;

                (E1 . i) in ( rng E1) by A21, FUNCT_1: 3;

                then

                 A159: (E1 . i) in (( rng E1) \/ ( rng E2)) by XBOOLE_1: 7, TARSKI:def 3;

                 not (E1 . i) in ( rng ( omega -exponent A1))

                proof

                  assume (E1 . i) in ( rng ( omega -exponent A1));

                  then

                  consider k be object such that

                   A160: k in ( dom ( omega -exponent A1)) & (( omega -exponent A1) . k) = (E1 . i) by FUNCT_1:def 3;

                  k in ( dom A1) by A160, Def1;

                  then

                   A161: k in i & k in ( dom C1) by RELAT_1: 57;

                  

                   A162: (E1 . i) = ((( omega -exponent C1) | i) . k) by A160, Th48

                  .= (( omega -exponent C1) . k) by A161, FUNCT_1: 49

                  .= ( omega -exponent (C1 . k)) by A161, Def1;

                  

                   A163: ( omega -exponent (C1 . i)) in ( omega -exponent (C1 . k)) by A22, A161, ORDINAL5:def 11;

                  (E1 . i) c= (( omega -exponent C1) . i) by Th97;

                  then (E1 . i) c= ( omega -exponent (C1 . i)) by A22, Def1;

                  hence contradiction by A162, A163, ORDINAL1: 12;

                end;

                then (E1 . i) in ( rng ( omega -exponent B1)) by A28, A159, XBOOLE_0:def 5;

                then

                consider k be object such that

                 A164: k in ( dom ( omega -exponent B1)) & (( omega -exponent B1) . k) = (E1 . i) by FUNCT_1:def 3;

                reconsider k as Nat by A164;

                

                 A165: k in ( dom B1) by A164, Def1;

                

                then

                 A166: (E1 . i) = ( omega -exponent (B1 . k)) by A164, Def1

                .= ( omega -exponent (C1 . (k + i))) by A165, AFINSQ_2:def 2;

                per cases ;

                  suppose k = 0 ;

                  hence contradiction by A158, A166;

                end;

                  suppose 0 < k;

                  then ( 0 + i) < (k + i) by XREAL_1: 8;

                  then i in ( Segm (k + i)) by NAT_1: 44;

                  then

                   A167: i in (k + i);

                  k in ( Segm ( len B1)) by A165;

                  then k < ( len B1) by NAT_1: 44;

                  then k < (( len C1) -' i) by AFINSQ_2:def 2;

                  then

                   A168: (k + i) < ((( len C1) -' i) + i) by XREAL_1: 8;

                  i in ( Segm ( len C1)) by A22;

                  then i < ( len C1) by NAT_1: 44;

                  then (k + i) < ( len C1) by A168, XREAL_1: 235;

                  then (k + i) in ( Segm ( len C1)) by NAT_1: 44;

                  hence contradiction by A158, A166, A167, ORDINAL5:def 11;

                end;

              end;

            end;

            then ( omega -exponent (C1 . i)) c= c by A14, Def1;

            then

             A169: ( omega -exponent (C1 . i)) in c by ORDINAL1: 11, A136, XBOOLE_0:def 8;

             not c in ( rng ( omega -exponent A1))

            proof

              assume c in ( rng ( omega -exponent A1));

              then

              consider m be object such that

               A171: m in ( dom ( omega -exponent A1)) & (( omega -exponent A1) . m) = c by FUNCT_1:def 3;

              reconsider m as Nat by A171;

              

               A172: m in ( dom A1) by A171, Def1;

              then

               A173: m in ( dom C1) & m in i by RELAT_1: 57;

              c = ( omega -exponent (A1 . m)) by A171, A172, Def1

              .= ( omega -exponent (C1 . m)) by A172, FUNCT_1: 47

              .= (( omega -exponent C1) . m) by A173, Def1;

              then (E1 . m) c= c by Th97;

              then

               A175: ( omega -exponent (( CantorNF a) . m)) c= c by Def1, A14, A173, ORDINAL1: 10;

              m in ( Segm i) by A173;

              then c in ( omega -exponent (( CantorNF a) . m)) by A14, NAT_1: 44;

              then c in c by A175;

              hence contradiction;

            end;

            then c in ( rng ( omega -exponent B1)) by A134, A28, XBOOLE_0:def 5;

            then

            consider m be object such that

             A176: m in ( dom ( omega -exponent B1)) & (( omega -exponent B1) . m) = c by FUNCT_1:def 3;

            reconsider m as Nat by A176;

            

             A177: m in ( dom B1) by A176, Def1;

            

            then

             A178: c = ( omega -exponent (B1 . m)) by A176, Def1

            .= ( omega -exponent (C1 . (m + i))) by A177, AFINSQ_2:def 2;

            per cases ;

              suppose m = 0 ;

              hence contradiction by A169, A178;

            end;

              suppose m > 0 ;

              then ( 0 + i) < (m + i) by XREAL_1: 8;

              then i in ( Segm (m + i)) by NAT_1: 44;

              then

               A179: i in (m + i);

              m in ( Segm ( len B1)) by A177;

              then m < ( len B1) by NAT_1: 44;

              then m < (( len C1) -' i) by AFINSQ_2:def 2;

              then

               A180: (m + i) < ((( len C1) -' i) + i) by XREAL_1: 8;

              i in ( Segm ( len C1)) by A22;

              then i < ( len C1) by NAT_1: 44;

              then (m + i) < ( len C1) by A180, XREAL_1: 235;

              then (m + i) in ( Segm ( len C1)) by NAT_1: 44;

              hence contradiction by A169, A178, A179, ORDINAL5:def 11;

            end;

          end;

            suppose

             A181: b = {} ;

            then

             A182: c = 0 by A132, ORDINAL5:def 10;

            assume ( omega -exponent (C1 . i)) <> c;

            hence contradiction by A14, A182, A181, Th82;

          end;

        end;

        

         A183: ( Sum^ B1) in (C2 . i)

        proof

          per cases by ORDINAL1: 16;

            suppose

             A184: B1 <> {} & c c= ( omega -exponent (( CantorNF a) . i));

            then

            consider b0 be Cantor-component Ordinal, B0 be Cantor-normal-form Ordinal-Sequence such that

             A185: B1 = ( <%b0%> ^ B0) by ORDINAL5: 67;

            

             A186: ( omega -exponent (C2 . i)) = ( omega -exponent (( CantorNF a) . i)) by A14, A86, A184, XBOOLE_0:def 10

            .= (E1 . i) by A14, Def1;

            then ( omega -exponent (C2 . i)) in ( rng E1) by A21, FUNCT_1: 3;

            then ( omega -exponent (C2 . i)) in (( rng E1) /\ ( rng E3)) by A17, A86, XBOOLE_0:def 4;

            

            then ( omega -leading_coeff (C2 . i)) = ((L1 . ((E1 " ) . ( omega -exponent (C2 . i)))) + (L3 . ((E3 " ) . ( omega -exponent (C2 . i))))) by A22, Th80

            .= ((L1 . i) + (L3 . ((E3 " ) . c))) by A21, A86, A186, FUNCT_1: 34

            .= ((L1 . i) + (L3 . 0 )) by A16, A20, FUNCT_1: 34;

            

            then

             A188: (C2 . i) = (((L1 . i) + (L3 . 0 )) *^ ( exp ( omega ,c))) by A22, A86, Th64

            .= (((L1 . i) +^ (L3 . 0 )) *^ ( exp ( omega ,c))) by CARD_2: 36

            .= (((L1 . i) *^ ( exp ( omega ,c))) +^ ((L3 . 0 ) *^ ( exp ( omega ,c)))) by ORDINAL3: 46;

             0 in ( dom L3) by A15, Def3;

            then (L3 . 0 ) <> {} by FUNCT_1:def 9;

            then

             A189: 0 in (L3 . 0 ) by XBOOLE_1: 61, ORDINAL1: 11;

            per cases by ORDINAL1: 16;

              suppose

               A190: ( omega -exponent b) in c;

              c c= (( omega -exponent C1) . i) by A86, A186, Th97;

              then

               A191: c c= ( omega -exponent (C1 . i)) by A22, Def1;

              

               A192: not ( omega -exponent (C1 . i)) in ( rng E2)

              proof

                assume ( omega -exponent (C1 . i)) in ( rng E2);

                then

                consider j be object such that

                 A193: j in ( dom E2) & (E2 . j) = ( omega -exponent (C1 . i)) by FUNCT_1:def 3;

                reconsider j as Nat by A193;

                

                 A194: j in ( dom ( CantorNF b)) by A193, Def1;

                per cases ;

                  suppose j = 0 ;

                  

                  then ( omega -exponent (C1 . i)) = ( omega -exponent (( CantorNF b) . 0 )) by A193, A194, Def1

                  .= ( omega -exponent ( Sum^ ( CantorNF b))) by Th44

                  .= ( omega -exponent b);

                  hence contradiction by A190, A191, ORDINAL1: 12;

                end;

                  suppose 0 < j;

                  then 0 in ( Segm j) by NAT_1: 44;

                  then ( omega -exponent (( CantorNF b) . j)) in ( omega -exponent (( CantorNF b) . 0 )) by A194, ORDINAL5:def 11;

                  then ( omega -exponent (C1 . i)) in ( omega -exponent (( CantorNF b) . 0 )) by A193, A194, Def1;

                  then ( omega -exponent (C1 . i)) in ( omega -exponent ( Sum^ ( CantorNF b))) by Th44;

                  hence contradiction by A190, A191;

                end;

              end;

              i in ( dom ( omega -exponent C1)) by A22, Def1;

              then (( omega -exponent C1) . i) in ( rng ( omega -exponent C1)) by FUNCT_1: 3;

              then ( omega -exponent (C1 . i)) in ( rng ( omega -exponent C1)) by A22, Def1;

              then ( omega -exponent (C1 . i)) in (( rng E1) \/ ( rng E2)) by Th76;

              then

               A195: ( omega -exponent (C1 . i)) in ( rng E1) by A192, XBOOLE_0:def 3;

              then

              consider j be object such that

               A196: j in ( dom E1) & (E1 . j) = ( omega -exponent (C1 . i)) by FUNCT_1:def 3;

              reconsider j as Nat by A196;

              

               A197: j in ( dom ( CantorNF a)) by A196, Def1;

              

               A198: i = j

              proof

                assume i <> j;

                per cases by XXREAL_0: 1;

                  suppose i < j;

                  then i in ( Segm j) by NAT_1: 44;

                  then ( omega -exponent (( CantorNF a) . j)) in ( omega -exponent (( CantorNF a) . i)) by A197, ORDINAL5:def 11;

                  then (E1 . j) in ( omega -exponent (( CantorNF a) . i)) by A197, Def1;

                  then ( omega -exponent (C1 . i)) in (E1 . i) by A14, A196, Def1;

                  then (( omega -exponent C1) . i) in (E1 . i) by A22, Def1;

                  then (( omega -exponent C1) . i) in (( omega -exponent C1) . i) by Th97, TARSKI:def 3;

                  hence contradiction;

                end;

                  suppose j < i;

                  then j in ( Segm i) by NAT_1: 44;

                  then

                   A199: j in i;

                  j in ( dom C1) by A197, Th77, TARSKI:def 3;

                  

                  then (( CantorNF a) . j) = (A1 . j) by A33, A199, RELAT_1: 57

                  .= (C1 . j) by A199, FUNCT_1: 49;

                  then

                   A200: ( omega -exponent (C1 . j)) = ( omega -exponent (C1 . i)) by A196, A197, Def1;

                  ( omega -exponent (C1 . i)) in ( omega -exponent (C1 . j)) by A22, A199, ORDINAL5:def 11;

                  hence contradiction by A200;

                end;

              end;

              

              then

               A201: ( omega -exponent (C1 . i)) = ( omega -exponent (( CantorNF a) . i)) by A14, A196, Def1

              .= c by A14, A184, XBOOLE_0:def 10;

              ( omega -exponent (C1 . i)) in (( rng E1) \ ( rng E2)) by A192, A195, XBOOLE_0:def 5;

              

              then ( omega -leading_coeff (C1 . i)) = (L1 . ((E1 " ) . ( omega -exponent (C1 . i)))) by A22, Th78

              .= (L1 . i) by A196, A198, FUNCT_1: 34;

              then

               A202: (C1 . i) = ((L1 . i) *^ ( exp ( omega ,c))) by A22, A201, Th64;

              

               A203: 0 in ( dom B1) by A184, XBOOLE_1: 61, ORDINAL1: 11;

              

               A204: b0 = (B1 . 0 ) by A185, AFINSQ_1: 35

              .= (C1 . ( 0 + i)) by A203, AFINSQ_2:def 2;

              then (b0 +^ ( Sum^ B0)) in (b0 +^ ( exp ( omega ,c))) by ORDINAL2: 32, A185, A201, Th43;

              then

               A205: ( Sum^ B1) in (((L1 . i) *^ ( exp ( omega ,c))) +^ ( exp ( omega ,c))) by A202, A185, A204, ORDINAL5: 55;

              1 c= (L3 . 0 ) by A189, CARD_1: 49, ZFMISC_1: 31;

              then (1 *^ ( exp ( omega ,c))) c= ((L3 . 0 ) *^ ( exp ( omega ,c))) by ORDINAL2: 41;

              then ( exp ( omega ,c)) c= ((L3 . 0 ) *^ ( exp ( omega ,c))) by ORDINAL2: 39;

              then (((L1 . i) *^ ( exp ( omega ,c))) +^ ( exp ( omega ,c))) c= (((L1 . i) *^ ( exp ( omega ,c))) +^ ((L3 . 0 ) *^ ( exp ( omega ,c)))) by ORDINAL2: 33;

              hence ( Sum^ B1) in (C2 . i) by A188, A205;

            end;

              suppose c c= ( omega -exponent b);

              then

               A206: c = ( omega -exponent b) by A2, XBOOLE_0:def 10;

              then

               A207: ( omega -exponent (C1 . i)) = c by A131;

              

               A208: 0 in ( dom B1) by A184, XBOOLE_1: 61, ORDINAL1: 11;

              ( exp ( omega ,( omega -exponent b0))) = ( exp ( omega ,( omega -exponent (B1 . 0 )))) by A185, AFINSQ_1: 35

              .= ( exp ( omega ,( omega -exponent (C1 . ( 0 + i))))) by A208, AFINSQ_2:def 2

              .= (1 *^ ( exp ( omega ,c))) by A131, A206, ORDINAL2: 39;

              then

               A209: ( Sum^ B0) in (1 *^ ( exp ( omega ,c))) by A185, Th43;

              

               A210: ( omega -exponent (C1 . i)) = ( omega -exponent (( CantorNF a) . i)) by A14, A131, A184, A206, XBOOLE_0:def 10

              .= (E1 . i) by A14, Def1;

              then

               A211: ( omega -exponent (C1 . i)) in ( rng E1) by A21, FUNCT_1: 3;

              per cases ;

                suppose b <> {} ;

                then

                 A212: 0 in ( dom ( CantorNF b)) by XBOOLE_1: 61, ORDINAL1: 11;

                then

                 A213: 0 in ( dom E2) by Def1;

                then (E2 . 0 ) in ( rng E2) by FUNCT_1: 3;

                then ( omega -exponent (( CantorNF b) . 0 )) in ( rng E2) by A212, Def1;

                then ( omega -exponent ( Sum^ ( CantorNF b))) in ( rng E2) by Th44;

                then

                 A214: c in ( rng E2) by A206;

                

                 A215: c = ( omega -exponent ( Sum^ ( CantorNF b))) by A206

                .= ( omega -exponent (( CantorNF b) . 0 )) by Th44

                .= (E2 . 0 ) by A212, Def1;

                ( omega -exponent (C1 . i)) in (( rng E1) /\ ( rng E2)) by A131, A206, A211, A214, XBOOLE_0:def 4;

                

                then ( omega -leading_coeff (C1 . i)) = ((L1 . ((E1 " ) . ( omega -exponent (C1 . i)))) + (L2 . ((E2 " ) . ( omega -exponent (C1 . i))))) by A22, Th80

                .= ((L1 . i) + (L2 . ((E2 " ) . c))) by A21, A131, A206, A210, FUNCT_1: 34

                .= ((L1 . i) + (L2 . 0 )) by A213, A215, FUNCT_1: 34;

                

                then

                 A216: (C1 . i) = (((L1 . i) + (L2 . 0 )) *^ ( exp ( omega ,( omega -exponent (C1 . i))))) by A22, Th64

                .= (((L1 . i) +^ (L2 . 0 )) *^ ( exp ( omega ,c))) by A131, A206, CARD_2: 36

                .= (((L1 . i) *^ ( exp ( omega ,c))) +^ ((L2 . 0 ) *^ ( exp ( omega ,c)))) by ORDINAL3: 46;

                (L2 . 0 ) in (L3 . 0 )

                proof

                  assume not (L2 . 0 ) in (L3 . 0 );

                  then

                   A217: (L3 . 0 ) c= (L2 . 0 ) by ORDINAL1: 16;

                  then 0 in ( dom L2) by FUNCT_1:def 2, A189;

                  then

                   A218: 0 in ( dom ( CantorNF b)) by Def3;

                  (L3 . 0 ) in (L2 . 0 )

                  proof

                    assume not (L3 . 0 ) in (L2 . 0 );

                    then (L2 . 0 ) c= (L3 . 0 ) by ORDINAL1: 16;

                    then (L2 . 0 ) = (L3 . 0 ) by A217, XBOOLE_0:def 10;

                    

                    then (( CantorNF d) . 0 ) = ((L2 . 0 ) *^ ( exp ( omega ,c))) by A15, A20, Th65

                    .= (( CantorNF b) . 0 ) by A215, A218, Th65;

                    hence contradiction by A1;

                  end;

                  then (L3 . 0 ) in ( Segm (L2 . 0 ));

                  then (L3 . 0 ) < (L2 . 0 ) by NAT_1: 44;

                  then ((L3 . 0 ) + 1) <= (L2 . 0 ) by NAT_1: 13;

                  then ( Segm ((L3 . 0 ) + 1)) c= ( Segm (L2 . 0 )) by NAT_1: 39;

                  then (((L3 . 0 ) + 1) *^ ( exp ( omega ,c))) c= ((L2 . 0 ) *^ ( exp ( omega ,c))) by ORDINAL2: 41;

                  then (((L3 . 0 ) + 1) *^ ( exp ( omega ,c))) c= (( CantorNF b) . 0 ) by A215, A218, Th65;

                  then (((L3 . 0 ) +^ 1) *^ ( exp ( omega ,c))) c= (( CantorNF b) . 0 ) by CARD_2: 36;

                  then (((L3 . 0 ) *^ ( exp ( omega ,c))) +^ (1 *^ ( exp ( omega ,c)))) c= (( CantorNF b) . 0 ) by ORDINAL3: 46;

                  then

                   A220: ((( CantorNF d) . 0 ) +^ (1 *^ ( exp ( omega ,c)))) c= (( CantorNF b) . 0 ) by A15, A20, Th65;

                  consider d0 be Cantor-component Ordinal, D0 be Cantor-normal-form Ordinal-Sequence such that

                   A221: ( CantorNF d) = ( <%d0%> ^ D0) by A1, ORDINAL5: 67;

                  ( exp ( omega ,( omega -exponent d0))) = ( exp ( omega ,( omega -exponent (( CantorNF d) . 0 )))) by A221, AFINSQ_1: 35

                  .= ( exp ( omega ,(E3 . 0 ))) by A15, Def1

                  .= (1 *^ ( exp ( omega ,c))) by A20, ORDINAL2: 39;

                  then ((( CantorNF d) . 0 ) +^ ( Sum^ D0)) in ((( CantorNF d) . 0 ) +^ (1 *^ ( exp ( omega ,c)))) by A221, Th43, ORDINAL2: 32;

                  then ((( CantorNF d) . 0 ) +^ ( Sum^ D0)) in (( CantorNF b) . 0 ) by A220;

                  then (d0 +^ ( Sum^ D0)) in (( CantorNF b) . 0 ) by A221, AFINSQ_1: 35;

                  then ( Sum^ ( CantorNF d)) in (( CantorNF b) . 0 ) by A221, ORDINAL5: 55;

                  then d in ( Sum^ ( CantorNF b)) by ORDINAL5: 56, TARSKI:def 3;

                  hence contradiction by A1;

                end;

                then (L2 . 0 ) in ( Segm (L3 . 0 ));

                then (L2 . 0 ) < (L3 . 0 ) by NAT_1: 44;

                then ((L2 . 0 ) + 1) <= (L3 . 0 ) by NAT_1: 13;

                then ( Segm ((L2 . 0 ) + 1)) c= ( Segm (L3 . 0 )) by NAT_1: 39;

                then (((L2 . 0 ) + 1) *^ ( exp ( omega ,c))) c= ((L3 . 0 ) *^ ( exp ( omega ,c))) by ORDINAL2: 41;

                then (((L2 . 0 ) +^ 1) *^ ( exp ( omega ,c))) c= ((L3 . 0 ) *^ ( exp ( omega ,c))) by CARD_2: 36;

                then (((L2 . 0 ) *^ ( exp ( omega ,c))) +^ (1 *^ ( exp ( omega ,c)))) c= ((L3 . 0 ) *^ ( exp ( omega ,c))) by ORDINAL3: 46;

                then (((L1 . i) *^ ( exp ( omega ,c))) +^ (((L2 . 0 ) *^ ( exp ( omega ,c))) +^ (1 *^ ( exp ( omega ,c))))) c= (C2 . i) by A188, ORDINAL2: 33;

                then

                 A222: ((C1 . i) +^ (1 *^ ( exp ( omega ,c)))) c= (C2 . i) by A216, ORDINAL3: 30;

                ((C1 . i) +^ ( Sum^ B0)) in ((C1 . i) +^ (1 *^ ( exp ( omega ,c)))) by A209, ORDINAL2: 32;

                then ((C1 . ( 0 + i)) +^ ( Sum^ B0)) in (C2 . i) by A222;

                then ((B1 . 0 ) +^ ( Sum^ B0)) in (C2 . i) by A208, AFINSQ_2:def 2;

                then (b0 +^ ( Sum^ B0)) in (C2 . i) by A185, AFINSQ_1: 35;

                hence ( Sum^ B1) in (C2 . i) by A185, ORDINAL5: 55;

              end;

                suppose b = {} ;

                then

                 A223: not ( omega -exponent (C1 . i)) in ( rng E2);

                i in ( dom ( omega -exponent C1)) by A22, Def1;

                then (( omega -exponent C1) . i) in ( rng ( omega -exponent C1)) by FUNCT_1: 3;

                then (( omega -exponent C1) . i) in (( rng E1) \/ ( rng E2)) by Th76;

                then ( omega -exponent (C1 . i)) in (( rng E1) \/ ( rng E2)) by A22, Def1;

                then ( omega -exponent (C1 . i)) in ( rng E1) or ( omega -exponent (C1 . i)) in ( rng E2) by XBOOLE_0:def 3;

                then ( omega -exponent (C1 . i)) in (( rng E1) \ ( rng E2)) by A223, XBOOLE_0:def 5;

                

                then

                 A224: ( omega -leading_coeff (C1 . i)) = (L1 . ((E1 " ) . ( omega -exponent (C1 . i)))) by A22, Th78

                .= (L1 . i) by A21, A210, FUNCT_1: 34;

                1 c= (L3 . 0 ) by A189, ZFMISC_1: 31, CARD_1: 49;

                then (1 *^ ( exp ( omega ,c))) c= ((L3 . 0 ) *^ ( exp ( omega ,c))) by ORDINAL2: 41;

                then (((L1 . i) *^ ( exp ( omega ,c))) +^ ( Sum^ B0)) in (C2 . i) by A209, A188, ORDINAL2: 32;

                then ((C1 . ( 0 + i)) +^ ( Sum^ B0)) in (C2 . i) by A22, A207, A224, Th64;

                then ((B1 . 0 ) +^ ( Sum^ B0)) in (C2 . i) by A208, AFINSQ_2:def 2;

                then (b0 +^ ( Sum^ B0)) in (C2 . i) by A185, AFINSQ_1: 35;

                hence ( Sum^ B1) in (C2 . i) by A185, ORDINAL5: 55;

              end;

            end;

          end;

            suppose

             A225: B1 <> {} & ( omega -exponent (( CantorNF a) . i)) in c;

            

             A226: (C2 . i) is Cantor-component by A22, ORDINAL5:def 11;

            per cases by ORDINAL1: 16;

              suppose

               A227: ( omega -exponent b) in c;

              

               A228: ( omega -exponent (C1 . i)) in c

              proof

                assume not ( omega -exponent (C1 . i)) in c;

                then

                 A229: c c= ( omega -exponent (C1 . i)) by ORDINAL1: 16;

                i in ( dom ( omega -exponent C1)) by A22, Def1;

                then (( omega -exponent C1) . i) in ( rng ( omega -exponent C1)) by FUNCT_1: 3;

                then ( omega -exponent (C1 . i)) in ( rng ( omega -exponent C1)) by A22, Def1;

                then ( omega -exponent (C1 . i)) in (( rng E1) \/ ( rng E2)) by Th76;

                per cases by XBOOLE_0:def 3;

                  suppose ( omega -exponent (C1 . i)) in ( rng E1);

                  then

                  consider j be object such that

                   A230: j in ( dom E1) & (E1 . j) = ( omega -exponent (C1 . i)) by FUNCT_1:def 3;

                  reconsider j as Nat by A230;

                  

                   A231: j in ( dom ( CantorNF a)) by A230, Def1;

                  then

                   A232: j in ( dom C1) by Th77, TARSKI:def 3;

                  

                   A233: ( omega -exponent (C1 . i)) = ( omega -exponent (( CantorNF a) . j)) by A230, A231, Def1;

                  per cases by ORDINAL1: 14;

                    suppose

                     A234: j in i;

                    

                    then (( CantorNF a) . j) = (A1 . j) by A33, A232, RELAT_1: 57

                    .= (C1 . j) by A234, FUNCT_1: 49;

                    then

                     A235: ( omega -exponent (C1 . i)) = ( omega -exponent (C1 . j)) by A233;

                    ( omega -exponent (C1 . i)) in ( omega -exponent (C1 . j)) by A22, A234, ORDINAL5:def 11;

                    hence contradiction by A235;

                  end;

                    suppose j = i;

                    hence contradiction by A225, A229, A233, ORDINAL1: 12;

                  end;

                    suppose i in j;

                    then ( omega -exponent (( CantorNF a) . j)) in ( omega -exponent (( CantorNF a) . i)) by A231, ORDINAL5:def 11;

                    hence contradiction by A229, A233, A225;

                  end;

                end;

                  suppose ( omega -exponent (C1 . i)) in ( rng E2);

                  then

                  consider j be object such that

                   A236: j in ( dom E2) & (E2 . j) = ( omega -exponent (C1 . i)) by FUNCT_1:def 3;

                  reconsider j as Nat by A236;

                  

                   A237: j in ( dom ( CantorNF b)) by A236, Def1;

                  then

                   A238: ( omega -exponent (C1 . i)) = ( omega -exponent (( CantorNF b) . j)) by A236, Def1;

                  per cases ;

                    suppose j = 0 ;

                    

                    then ( omega -exponent (C1 . i)) = ( omega -exponent ( Sum^ ( CantorNF b))) by A238, Th44

                    .= ( omega -exponent b);

                    hence contradiction by A227, A229, ORDINAL1: 12;

                  end;

                    suppose 0 < j;

                    then 0 in ( Segm j) by NAT_1: 44;

                    then ( omega -exponent (C1 . i)) in ( omega -exponent (( CantorNF b) . 0 )) by A237, A238, ORDINAL5:def 11;

                    then ( omega -exponent (C1 . i)) in ( omega -exponent ( Sum^ ( CantorNF b))) by Th44;

                    hence contradiction by A227, A229;

                  end;

                end;

              end;

              now

                let j be Ordinal;

                assume

                 A239: j in ( dom B1);

                then

                reconsider m = j as Nat;

                

                 A240: (B1 . j) is Cantor-component by A239, ORDINAL5:def 11;

                per cases ;

                  suppose m = 0 ;

                  then ( omega -exponent (B1 . m)) = ( omega -exponent (C1 . ( 0 + i))) by A239, AFINSQ_2:def 2;

                  then ( exp ( omega ,( omega -exponent (B1 . j)))) in ( exp ( omega ,c)) by A228, ORDINAL4: 24;

                  then (( omega -leading_coeff (B1 . j)) *^ ( exp ( omega ,( omega -exponent (B1 . j))))) in ( exp ( omega ,c)) by A240, Th42;

                  hence (B1 . j) in ( exp ( omega ,c)) by A240, Th59;

                end;

                  suppose 0 < m;

                  then 0 in ( Segm j) by NAT_1: 44;

                  then

                   A242: 0 in j;

                   0 in ( dom B1) by A239, XBOOLE_1: 61, ORDINAL1: 11;

                  then (B1 . 0 ) = (C1 . ( 0 + i)) by AFINSQ_2:def 2;

                  then ( omega -exponent (B1 . j)) in ( omega -exponent (C1 . i)) by A239, A242, ORDINAL5:def 11;

                  then ( omega -exponent (B1 . j)) in c by A228, ORDINAL1: 10;

                  then ( exp ( omega ,( omega -exponent (B1 . j)))) in ( exp ( omega ,c)) by ORDINAL4: 24;

                  then (( omega -leading_coeff (B1 . j)) *^ ( exp ( omega ,( omega -exponent (B1 . j))))) in ( exp ( omega ,c)) by A240, Th42;

                  hence (B1 . j) in ( exp ( omega ,c)) by A240, Th59;

                end;

              end;

              then ( Sum^ B1) in ( exp ( omega ,( omega -exponent (C2 . i)))) by A86, Th41;

              then ( Sum^ B1) in (( omega -leading_coeff (C2 . i)) *^ ( exp ( omega ,( omega -exponent (C2 . i))))) by A226, ORDINAL3: 32;

              hence ( Sum^ B1) in (C2 . i) by A226, Th59;

            end;

              suppose c c= ( omega -exponent b);

              then

               A243: ( omega -exponent b) = c by A2, XBOOLE_0:def 10;

              then

               A244: ( omega -exponent (C1 . i)) = c by A131;

              

               A245: not c in ( rng E1)

              proof

                assume c in ( rng E1);

                then

                consider j be object such that

                 A246: j in ( dom E1) & (E1 . j) = c by FUNCT_1:def 3;

                reconsider j as Nat by A246;

                

                 A247: j in ( dom ( CantorNF a)) by A246, Def1;

                then

                 A248: ( omega -exponent (( CantorNF a) . j)) = c by A246, Def1;

                per cases by ORDINAL1: 14;

                  suppose i in j;

                  hence contradiction by A225, A247, A248, ORDINAL5:def 11;

                end;

                  suppose i = j;

                  hence contradiction by A225, A248;

                end;

                  suppose j in i;

                  then j in ( Segm i);

                  then c in ( omega -exponent (( CantorNF a) . j)) by A14, NAT_1: 44;

                  hence contradiction by A248;

                end;

              end;

              i in ( dom ( omega -exponent C1)) by A22, Def1;

              then (( omega -exponent C1) . i) in ( rng ( omega -exponent C1)) by FUNCT_1: 3;

              then ( omega -exponent (C1 . i)) in ( rng ( omega -exponent C1)) by A22, Def1;

              then

               A249: ( omega -exponent (C1 . i)) in (( rng E1) \/ ( rng E2)) by Th76;

              then b <> {} by A244, A245, XBOOLE_0:def 3;

              then

               A250: 0 in ( dom ( CantorNF b)) by XBOOLE_1: 61, ORDINAL1: 11;

              then

               A251: 0 in ( dom E2) by Def1;

              

               A252: c = ( omega -exponent ( Sum^ ( CantorNF b))) by A243

              .= ( omega -exponent (( CantorNF b) . 0 )) by Th44

              .= (E2 . 0 ) by A250, Def1;

              ( omega -exponent (C1 . i)) in ( rng E1) or ( omega -exponent (C1 . i)) in ( rng E2) by A249, XBOOLE_0:def 3;

              then ( omega -exponent (C1 . i)) in (( rng E2) \ ( rng E1)) by A244, A245, XBOOLE_0:def 5;

              

              then

               A253: ( omega -leading_coeff (C1 . i)) = (L2 . ((E2 " ) . (E2 . 0 ))) by A22, A244, A252, Th79

              .= (L2 . 0 ) by A251, FUNCT_1: 34;

              ( omega -exponent (C2 . i)) in ( rng E1) or ( omega -exponent (C2 . i)) in ( rng E3) by A18, A86, XBOOLE_0:def 3;

              then ( omega -exponent (C2 . i)) in (( rng E3) \ ( rng E1)) by A86, A245, XBOOLE_0:def 5;

              

              then

               A254: ( omega -leading_coeff (C2 . i)) = (L3 . ((E3 " ) . (E3 . 0 ))) by A20, A22, A86, Th79

              .= (L3 . 0 ) by A16, FUNCT_1: 34;

              (L2 . 0 ) in (L3 . 0 )

              proof

                assume not (L2 . 0 ) in (L3 . 0 );

                then

                 A255: (L3 . 0 ) c= (L2 . 0 ) by ORDINAL1: 16;

                (L3 . 0 ) in (L2 . 0 )

                proof

                  assume not (L3 . 0 ) in (L2 . 0 );

                  then (L2 . 0 ) c= (L3 . 0 ) by ORDINAL1: 16;

                  then

                   A256: (L2 . 0 ) = (L3 . 0 ) by A255, XBOOLE_0:def 10;

                  (( CantorNF b) . 0 ) = ((L2 . 0 ) *^ ( exp ( omega ,(E2 . 0 )))) by A250, Th65

                  .= (( CantorNF d) . 0 ) by A15, A20, A252, A256, Th65;

                  hence contradiction by A1;

                end;

                then (L3 . 0 ) in ( Segm (L2 . 0 ));

                then (L3 . 0 ) < (L2 . 0 ) by NAT_1: 44;

                then ((L3 . 0 ) + 1) <= (L2 . 0 ) by NAT_1: 13;

                then ( Segm ((L3 . 0 ) + 1)) c= ( Segm (L2 . 0 )) by NAT_1: 39;

                then (((L3 . 0 ) + 1) *^ ( exp ( omega ,c))) c= ((L2 . 0 ) *^ ( exp ( omega ,c))) by ORDINAL2: 41;

                then (((L3 . 0 ) + 1) *^ ( exp ( omega ,c))) c= (( CantorNF b) . 0 ) by A250, A252, Th65;

                then (((L3 . 0 ) +^ 1) *^ ( exp ( omega ,c))) c= (( CantorNF b) . 0 ) by CARD_2: 36;

                then

                 A257: (((L3 . 0 ) *^ ( exp ( omega ,c))) +^ (1 *^ ( exp ( omega ,c)))) c= (( CantorNF b) . 0 ) by ORDINAL3: 46;

                consider d0 be Cantor-component Ordinal, D0 be Cantor-normal-form Ordinal-Sequence such that

                 A258: ( CantorNF d) = ( <%d0%> ^ D0) by A1, ORDINAL5: 67;

                ( exp ( omega ,( omega -exponent d0))) = ( exp ( omega ,( omega -exponent (( CantorNF d) . 0 )))) by A258, AFINSQ_1: 35

                .= ( exp ( omega ,(E3 . 0 ))) by A15, Def1

                .= (1 *^ ( exp ( omega ,c))) by A20, ORDINAL2: 39;

                then (((L3 . 0 ) *^ ( exp ( omega ,c))) +^ ( Sum^ D0)) in (((L3 . 0 ) *^ ( exp ( omega ,c))) +^ (1 *^ ( exp ( omega ,c)))) by ORDINAL2: 32, A258, Th43;

                then (((L3 . 0 ) *^ ( exp ( omega ,c))) +^ ( Sum^ D0)) in (( CantorNF b) . 0 ) by A257;

                then ((( CantorNF d) . 0 ) +^ ( Sum^ D0)) in (( CantorNF b) . 0 ) by A15, A20, Th65;

                then ((( CantorNF d) . 0 ) +^ ( Sum^ D0)) in ( Sum^ ( CantorNF b)) by ORDINAL5: 56, TARSKI:def 3;

                then (d0 +^ ( Sum^ D0)) in b by A258, AFINSQ_1: 35;

                then ( Sum^ ( CantorNF d)) in b by A258, ORDINAL5: 55;

                hence contradiction by A1;

              end;

              then (L2 . 0 ) in ( Segm (L3 . 0 ));

              then (L2 . 0 ) < (L3 . 0 ) by NAT_1: 44;

              then ((L2 . 0 ) + 1) <= (L3 . 0 ) by NAT_1: 13;

              then ( Segm ((L2 . 0 ) + 1)) c= ( Segm (L3 . 0 )) by NAT_1: 39;

              then (((L2 . 0 ) + 1) *^ ( exp ( omega ,c))) c= ((L3 . 0 ) *^ ( exp ( omega ,c))) by ORDINAL2: 41;

              then (((L2 . 0 ) +^ 1) *^ ( exp ( omega ,c))) c= ((L3 . 0 ) *^ ( exp ( omega ,c))) by CARD_2: 36;

              then

               A259: (((L2 . 0 ) *^ ( exp ( omega ,c))) +^ (1 *^ ( exp ( omega ,c)))) c= ((L3 . 0 ) *^ ( exp ( omega ,c))) by ORDINAL3: 46;

              (C2 . i) = ((L3 . 0 ) *^ ( exp ( omega ,c))) by A22, A86, A254, Th64;

              then

               A260: (((L2 . 0 ) *^ ( exp ( omega ,c))) +^ (1 *^ ( exp ( omega ,c)))) c= (C2 . i) by A259;

              

               A261: (C1 . i) = ((L2 . 0 ) *^ ( exp ( omega ,c))) by A22, A244, A253, Th64;

              consider b0 be Cantor-component Ordinal, B0 be Cantor-normal-form Ordinal-Sequence such that

               A262: B1 = ( <%b0%> ^ B0) by A225, ORDINAL5: 67;

              

               A263: 0 in ( dom B1) by A225, XBOOLE_1: 61, ORDINAL1: 11;

              ( exp ( omega ,( omega -exponent b0))) = ( exp ( omega ,( omega -exponent (B1 . 0 )))) by A262, AFINSQ_1: 35

              .= ( exp ( omega ,( omega -exponent (C1 . ( 0 + i))))) by A263, AFINSQ_2:def 2

              .= (1 *^ ( exp ( omega ,c))) by A244, ORDINAL2: 39;

              then (((L2 . 0 ) *^ ( exp ( omega ,c))) +^ ( Sum^ B0)) in (((L2 . 0 ) *^ ( exp ( omega ,c))) +^ (1 *^ ( exp ( omega ,c)))) by A262, Th43, ORDINAL2: 32;

              then ((C1 . ( 0 + i)) +^ ( Sum^ B0)) in (C2 . i) by A260, A261;

              then ((B1 . 0 ) +^ ( Sum^ B0)) in (C2 . i) by A263, AFINSQ_2:def 2;

              then (b0 +^ ( Sum^ B0)) in (C2 . i) by A262, AFINSQ_1: 35;

              hence ( Sum^ B1) in (C2 . i) by A262, ORDINAL5: 55;

            end;

          end;

            suppose B1 = {} ;

            hence thesis by A22, XBOOLE_1: 61, ORDINAL1: 11, ORDINAL5: 52;

          end;

        end;

        i in ( Segm ( dom C2)) by A14, Th77, TARSKI:def 3;

        then ( 0 + i) < ( len C2) by NAT_1: 44;

        then ( Sum^ B1) in (B2 . 0 ) by A183, AFINSQ_2: 8;

        then ( Sum^ B1) in ( Sum^ B2) by ORDINAL5: 56, TARSKI:def 3;

        then

         A265: (( Sum^ A1) +^ ( Sum^ B1)) in (( Sum^ A2) +^ ( Sum^ B2)) by A85, ORDINAL2: 32;

        

         A266: (a (+) b) = ( Sum^ (A1 ^ B1))

        .= (( Sum^ A1) +^ ( Sum^ B1)) by Th24;

        (a (+) d) = ( Sum^ (A2 ^ B2))

        .= (( Sum^ A2) +^ ( Sum^ B2)) by Th24;

        hence thesis by A265, A266;

      end;

    end;

    theorem :: ORDINAL7:94

    

     Th107: for a,b,c be Ordinal st b in c holds (a (+) b) in (a (+) c)

    proof

      let a,b,c be Ordinal;

      assume

       A1: b in c;

      per cases ;

        suppose a = 0 ;

        then (a (+) b) = b & (a (+) c) = c by Th82;

        hence thesis by A1;

      end;

        suppose

         A2: a <> 0 ;

        defpred P[ Nat] means (( CantorNF b) . $1) <> (( CantorNF c) . $1);

        

         A3: ex i be Nat st P[i]

        proof

          assume

           A4: for i be Nat holds not P[i];

          

           A5: ( dom ( CantorNF b)) = ( dom ( CantorNF c))

          proof

            assume ( dom ( CantorNF b)) <> ( dom ( CantorNF c));

            per cases by XBOOLE_0:def 10;

              suppose not ( dom ( CantorNF b)) c= ( dom ( CantorNF c));

              then

               A6: (( CantorNF b) . ( dom ( CantorNF c))) <> {} by ORDINAL1: 16, FUNCT_1:def 9;

               not ( dom ( CantorNF c)) in ( dom ( CantorNF c));

              then (( CantorNF c) . ( dom ( CantorNF c))) = {} by FUNCT_1:def 2;

              hence contradiction by A4, A6;

            end;

              suppose not ( dom ( CantorNF c)) c= ( dom ( CantorNF b));

              then

               A7: (( CantorNF c) . ( dom ( CantorNF b))) <> {} by ORDINAL1: 16, FUNCT_1:def 9;

               not ( dom ( CantorNF b)) in ( dom ( CantorNF b));

              then (( CantorNF b) . ( dom ( CantorNF b))) = {} by FUNCT_1:def 2;

              hence contradiction by A4, A7;

            end;

          end;

          for x be object st x in ( dom ( CantorNF b)) holds (( CantorNF b) . x) = (( CantorNF c) . x) by A4;

          then ( Sum^ ( CantorNF b)) = ( Sum^ ( CantorNF c)) by A5, FUNCT_1: 2;

          hence contradiction by A1;

        end;

        consider i be Nat such that

         A8: P[i] & for j be Nat st P[j] holds i <= j from NAT_1:sch 5( A3);

        set A1 = (( CantorNF b) | i), A2 = (( CantorNF c) | i);

        set B1 = (( CantorNF b) /^ i), B2 = (( CantorNF c) /^ i);

        

         A9: i c= ( dom ( CantorNF b)) & i c= ( dom ( CantorNF c))

        proof

          assume not (i c= ( dom ( CantorNF b)) & i c= ( dom ( CantorNF c)));

          per cases ;

            suppose

             A10: not i c= ( dom ( CantorNF b));

            then

            consider x be object such that

             A11: x in i & not x in ( dom ( CantorNF b)) by TARSKI:def 3;

            i in omega by ORDINAL1:def 12;

            then x in omega by A11, ORDINAL1: 10;

            then

            reconsider x as Nat;

            

             A12: (( CantorNF b) . x) = {} by A11, FUNCT_1:def 2;

            x in ( Segm i) by A11;

            then (( CantorNF b) . x) = (( CantorNF c) . x) by A8, NAT_1: 44;

            then ( dom ( CantorNF c)) c= x by A12, FUNCT_1:def 9, ORDINAL1: 16;

            then ( dom ( CantorNF c)) in i by A11, ORDINAL1: 12;

            then

             A13: (( CantorNF c) . i) = {} by FUNCT_1:def 2;

            ( dom ( CantorNF b)) in i by A10, ORDINAL1: 16;

            hence contradiction by A8, A13, FUNCT_1:def 2;

          end;

            suppose

             A14: not i c= ( dom ( CantorNF c));

            then

            consider x be object such that

             A15: x in i & not x in ( dom ( CantorNF c)) by TARSKI:def 3;

            i in omega by ORDINAL1:def 12;

            then x in omega by A15, ORDINAL1: 10;

            then

            reconsider x as Nat;

            

             A16: (( CantorNF c) . x) = {} by A15, FUNCT_1:def 2;

            x in ( Segm i) by A15;

            then (( CantorNF b) . x) = (( CantorNF c) . x) by A8, NAT_1: 44;

            then ( dom ( CantorNF b)) c= x by A16, FUNCT_1:def 9, ORDINAL1: 16;

            then ( dom ( CantorNF b)) in i by A15, ORDINAL1: 12;

            then

             A17: (( CantorNF b) . i) = {} by FUNCT_1:def 2;

            ( dom ( CantorNF c)) in i by A14, ORDINAL1: 16;

            hence contradiction by A8, A17, FUNCT_1:def 2;

          end;

        end;

        

         A18: ( dom A1) = (( dom ( CantorNF b)) /\ i) by RELAT_1: 61

        .= i by A9, XBOOLE_1: 28

        .= (( dom ( CantorNF c)) /\ i) by A9, XBOOLE_1: 28

        .= ( dom A2) by RELAT_1: 61;

        for x be object st x in ( dom A1) holds (A1 . x) = (A2 . x)

        proof

          let x be object;

          assume x in ( dom A1);

          then

           A19: x in i by RELAT_1: 57;

          i in omega by ORDINAL1:def 12;

          then x in omega by A19, ORDINAL1: 10;

          then

          reconsider m = x as Nat;

          m in ( Segm i) by A19;

          then

           A20: m < i by NAT_1: 44;

          

          thus (A1 . x) = (( CantorNF b) . m) by A19, FUNCT_1: 49

          .= (( CantorNF c) . m) by A8, A20

          .= (A2 . x) by A19, FUNCT_1: 49;

        end;

        then

         A21: A1 = A2 by A18, FUNCT_1: 2;

        

         A22: ( Sum^ B1) in ( Sum^ B2)

        proof

          ( Sum^ ( CantorNF b)) in ( Sum^ (A2 ^ B2)) by A1;

          then ( Sum^ (A1 ^ B1)) in (( Sum^ A2) +^ ( Sum^ B2)) by Th24;

          then (( Sum^ A1) +^ ( Sum^ B1)) in (( Sum^ A2) +^ ( Sum^ B2)) by Th24;

          hence thesis by A21, ORDINAL3: 22;

        end;

        

         A23: (A1 ^ B1) is Cantor-normal-form;

        

         A24: (A2 ^ B2) is Cantor-normal-form;

        

         A25: b = ( Sum^ (A1 ^ B1))

        .= (( Sum^ A1) +^ ( Sum^ B1)) by Th24

        .= (( Sum^ A1) (+) ( Sum^ B1)) by A23, Th84;

        

         A26: c = ( Sum^ (A2 ^ B2))

        .= (( Sum^ A2) +^ ( Sum^ B2)) by Th24

        .= (( Sum^ A2) (+) ( Sum^ B2)) by A24, Th84;

        

         A27: (a (+) ( Sum^ A1)) is non empty by A2;

        (B1 . 0 ) <> (B2 . 0 )

        proof

           0 in ( dom B1) or 0 in ( dom B2)

          proof

            assume not 0 in ( dom B1) & not 0 in ( dom B2);

            then ( dom B1) c= {} & ( dom B2) c= {} by ORDINAL1: 16;

            then B1 = {} & B2 = {} ;

            then (A1 ^ B1) = A1 & (A2 ^ B2) = A2;

            then

             A28: ( CantorNF b) = A1 & ( CantorNF c) = A2;

             not i in i;

            then not i in (( dom ( CantorNF b)) /\ i) & not i in (( dom ( CantorNF c)) /\ i) by XBOOLE_0:def 4;

            then

             A29: not i in ( dom A1) & not i in ( dom A2) by RELAT_1: 61;

            

            then (( CantorNF b) . i) = {} by A28, FUNCT_1:def 2

            .= (( CantorNF c) . i) by A28, A29, FUNCT_1:def 2;

            hence contradiction by A8;

          end;

          per cases ;

            suppose 0 in ( dom B1) & 0 in ( dom B2);

            then (B1 . 0 ) = (( CantorNF b) . ( 0 + i)) & (B2 . 0 ) = (( CantorNF c) . ( 0 + i)) by AFINSQ_2:def 2;

            hence thesis by A8;

          end;

            suppose 0 in ( dom B1) & not 0 in ( dom B2);

            then (B1 . 0 ) <> {} & (B2 . 0 ) = {} by FUNCT_1:def 9, FUNCT_1:def 2;

            hence thesis;

          end;

            suppose not 0 in ( dom B1) & 0 in ( dom B2);

            then (B1 . 0 ) = {} & (B2 . 0 ) <> {} by FUNCT_1:def 9, FUNCT_1:def 2;

            hence thesis;

          end;

        end;

        then (( CantorNF ( Sum^ B1)) . 0 ) <> (( CantorNF ( Sum^ B2)) . 0 );

        then ((a (+) ( Sum^ A1)) (+) ( Sum^ B1)) in ((a (+) ( Sum^ A2)) (+) ( Sum^ B2)) by A21, A22, A27, Lm11;

        then (a (+) b) in ((a (+) ( Sum^ A2)) (+) ( Sum^ B2)) by A25, Th81;

        hence thesis by A26, Th81;

      end;

    end;

    theorem :: ORDINAL7:95

    for a,b,c be Ordinal st b c= c holds (a (+) b) c= (a (+) c)

    proof

      let a,b,c be Ordinal;

      assume

       A1: b c= c;

      per cases by ORDINAL1: 16;

        suppose c c= b;

        hence thesis by A1, XBOOLE_0:def 10;

      end;

        suppose b in c;

        hence thesis by Th107, ORDINAL1:def 2;

      end;

    end;