measure9.miz



    begin

    theorem :: MEASURE9:1

    

     Th52: for K be Relation st ( rng K) is empty-membered holds ( union ( rng K)) = {}

    proof

      let K be Relation;

      assume

       A2: ( rng K) is empty-membered;

      now

        let x be object;

        assume x in ( union ( rng K));

        then ex A be set st x in A & A in ( rng K) by TARSKI:def 4;

        hence x in {} by A2;

      end;

      then ( union ( rng K)) c= {} by TARSKI:def 3;

      hence ( union ( rng K)) = {} ;

    end;

    theorem :: MEASURE9:2

    for K be Function holds ( rng K) is empty-membered iff (for x be object holds (K . x) = {} )

    proof

      let K be Function;

      hereby

        assume

         A1: ( rng K) is empty-membered;

        let x be object;

        per cases ;

          suppose x in ( dom K);

          hence (K . x) = {} by A1, FUNCT_1: 3;

        end;

          suppose not x in ( dom K);

          hence (K . x) = {} by FUNCT_1:def 2;

        end;

      end;

      assume

       A2: for x be object holds (K . x) = {} ;

      now

        assume ex y be non empty set st y in ( rng K);

        then

        consider y be non empty set such that

         A3: y in ( rng K);

        ex a be object st a in ( dom K) & y = (K . a) by A3, FUNCT_1:def 3;

        hence contradiction by A2;

      end;

      hence ( rng K) is empty-membered;

    end;

    definition

      let D be set, F be FinSequenceSet of D, f be FinSequence of F, n be Nat;

      :: original: .

      redefine

      func f . n -> FinSequence of D ;

      correctness

      proof

        per cases ;

          suppose n in ( dom f);

          then (f . n) in ( rng f) by FUNCT_1: 3;

          hence (f . n) is FinSequence of D by FINSEQ_2:def 3;

        end;

          suppose not n in ( dom f);

          then (f . n) = ( <*> D) by FUNCT_1:def 2;

          hence (f . n) is FinSequence of D;

        end;

      end;

    end

    definition

      let D be set, Y be FinSequenceSet of D, F be FinSequence of Y;

      :: MEASURE9:def1

      func Length F -> FinSequence of NAT means

      : Def1: ( dom it ) = ( dom F) & for n be Nat st n in ( dom it ) holds (it . n) = ( len (F . n));

      existence

      proof

        defpred P[ Nat, object] means $2 = ( len (F . $1));

        

         A1: for k be Nat st k in ( Seg ( len F)) holds ex x be Element of NAT st P[k, x];

        consider IT be FinSequence of NAT such that

         A2: ( dom IT) = ( Seg ( len F)) & for k be Nat st k in ( Seg ( len F)) holds P[k, (IT . k)] from FINSEQ_1:sch 5( A1);

        take IT;

        thus ( dom IT) = ( dom F) by A2, FINSEQ_1:def 3;

        thus for n be Nat st n in ( dom IT) holds (IT . n) = ( len (F . n)) by A2;

      end;

      uniqueness

      proof

        let IT1,IT2 be FinSequence of NAT ;

        assume that

         A1: ( dom IT1) = ( dom F) & for n be Nat st n in ( dom IT1) holds (IT1 . n) = ( len (F . n)) and

         A2: ( dom IT2) = ( dom F) & for n be Nat st n in ( dom IT2) holds (IT2 . n) = ( len (F . n));

        

         A3: ( len IT1) = ( len IT2) by A1, A2, FINSEQ_3: 29;

        now

          let k be Nat;

          assume k in ( dom IT1);

          then (IT1 . k) = ( len (F . k)) & (IT2 . k) = ( len (F . k)) by A1, A2;

          hence (IT1 . k) = (IT2 . k);

        end;

        hence IT1 = IT2 by A3, FINSEQ_2: 9;

      end;

    end

    theorem :: MEASURE9:3

    for D be set, Y be FinSequenceSet of D, F be FinSequence of Y st (for n be Nat st n in ( dom F) holds (F . n) = ( <*> D)) holds ( Sum ( Length F)) = 0

    proof

      let D be set, Y be FinSequenceSet of D, F be FinSequence of Y;

      assume

       A1: for n be Nat st n in ( dom F) holds (F . n) = ( <*> D);

      

       A2: ( dom ( Length F)) = ( dom F) by Def1

      .= ( Seg ( len F)) by FINSEQ_1:def 3;

      

       A6: (( len F) |-> 0 qua Real) = (( Seg ( len F)) --> 0 qua Real) by FINSEQ_2:def 2;

      then

       A3: ( dom (( len F) |-> 0 qua Real)) = ( Seg ( len F)) by FUNCT_2:def 1;

      now

        let k be Nat;

        assume

         A4: k in ( dom ( Length F));

        then k in ( dom F) by Def1;

        then (F . k) = ( <*> D) by A1;

        then (( Length F) . k) = 0 by A4, Def1;

        hence (( Length F) . k) = ((( len F) |-> 0 qua Real) . k) by A2, A4, A6, FUNCOP_1: 7;

      end;

      then ( Length F) = (( len F) |-> 0 qua Real) by A2, A3, FINSEQ_1: 13;

      hence ( Sum ( Length F)) = 0 by RVSUM_1: 81;

    end;

    theorem :: MEASURE9:4

    

     Th2: for D be set, Y be FinSequenceSet of D, F be FinSequence of Y, k be Nat st k < ( len F) holds ( Length (F | (k + 1))) = (( Length (F | k)) ^ <*( len (F . (k + 1)))*>)

    proof

      let D be set, Y be FinSequenceSet of D, F be FinSequence of Y, k be Nat;

      assume

       A1: k < ( len F);

      then (k + 1) <= ( len F) by NAT_1: 13;

      then

       A3: ( len (F | (k + 1))) = (k + 1) by FINSEQ_1: 59;

      

       A6: ( len (F | k)) = k by A1, FINSEQ_1: 59;

      

       A5: ( dom ( Length (F | (k + 1)))) = ( dom (F | (k + 1))) & ( dom ( Length (F | k))) = ( dom (F | k)) by Def1;

      then

       A7: ( len ( Length (F | (k + 1)))) = (k + 1) & ( len ( Length (F | k))) = k by A3, A6, FINSEQ_3: 29;

      

      then

       A8: ( len (( Length (F | k)) ^ <*( len (F . (k + 1)))*>)) = (k + ( len <*( len (F . (k + 1)))*>)) by FINSEQ_1: 22

      .= (k + 1) by FINSEQ_1: 40;

      now

        let n be Nat;

        assume

         A9: 1 <= n & n <= ( len ( Length (F | (k + 1))));

        then n in ( dom ( Length (F | (k + 1)))) by FINSEQ_3: 25;

        

        then

         A10: (( Length (F | (k + 1))) . n) = ( len ((F | (k + 1)) . n)) by Def1

        .= ( len (F . n)) by A7, A9, FINSEQ_3: 112;

        per cases ;

          suppose n = ( len ( Length (F | (k + 1))));

          hence (( Length (F | (k + 1))) . n) = ((( Length (F | k)) ^ <*( len (F . (k + 1)))*>) . n) by A7, A10, FINSEQ_1: 42;

        end;

          suppose n <> ( len ( Length (F | (k + 1))));

          then n < (k + 1) by A7, A9, XXREAL_0: 1;

          then

           A11: n <= k by NAT_1: 13;

          

          then ((( Length (F | k)) ^ <*( len (F . (k + 1)))*>) . n) = (( Length (F | k)) . n) by A9, A7, FINSEQ_1: 64

          .= ( len ((F | k) . n)) by A11, Def1, A9, A5, A6, FINSEQ_3: 25

          .= ( len (F . n)) by A11, FINSEQ_3: 112;

          hence (( Length (F | (k + 1))) . n) = ((( Length (F | k)) ^ <*( len (F . (k + 1)))*>) . n) by A10;

        end;

      end;

      hence thesis by A5, A8, A3, FINSEQ_3: 29;

    end;

    theorem :: MEASURE9:5

    

     Th3: for D be set, Y be FinSequenceSet of D, F be FinSequence of Y, n be Nat st 1 <= n & n <= ( Sum ( Length F)) holds ex k,m be Nat st 1 <= m & m <= ( len (F . (k + 1))) & k < ( len F) & (m + ( Sum ( Length (F | k)))) = n & n <= ( Sum ( Length (F | (k + 1))))

    proof

      let D be set, Y be FinSequenceSet of D, F be FinSequence of Y, n be Nat;

      assume

       A1: 1 <= n & n <= ( Sum ( Length F));

      now

        assume

         A2: for k be Nat holds n <= ( Sum ( Length (F | k))) or n > ( Sum ( Length (F | (k + 1))));

        defpred P[ Nat] means n > ( Sum ( Length (F | ($1 + 1))));

        ( dom ( Length (F | 0 ))) = ( dom {} ) by Def1;

        then ( Length (F | 0 )) = {} ;

        then

         A3: P[ 0 ] by A2, A1, RVSUM_1: 72;

        

         A4: for k be Nat st P[k] holds P[(k + 1)] by A2;

        for k be Nat holds P[k] from NAT_1:sch 2( A3, A4);

        then n > ( Sum ( Length (F | (( len F) + 1))));

        hence contradiction by A1, FINSEQ_1: 58, NAT_1: 11;

      end;

      then

      consider k be Nat such that

       A6: ( Sum ( Length (F | k))) < n & n <= ( Sum ( Length (F | (k + 1))));

      consider m be Nat such that

       A7: n = (( Sum ( Length (F | k))) + m) by A6, NAT_1: 10;

      take k, m;

       A8:

      now

        assume

         A9: ( len F) <= k;

        k <= (k + 1) by NAT_1: 11;

        then (F | (k + 1)) = F & (F | k) = F by A9, XXREAL_0: 2, FINSEQ_1: 58;

        hence contradiction by A6;

      end;

      then ( Length (F | (k + 1))) = (( Length (F | k)) ^ <*( len (F . (k + 1)))*>) by Th2;

      then (m + ( Sum ( Length (F | k)))) <= (( Sum ( Length (F | k))) + ( len (F . (k + 1)))) by A6, A7, RVSUM_1: 74;

      hence thesis by A6, A7, NAT_1: 19, A8, XREAL_1: 6;

    end;

    

     RFINSEQlm3: for n be Nat, D be set, f be FinSequence of D st ( len f) <= n holds (f | n) = f

    proof

      let n be Nat, D be set, f be FinSequence of D;

      

       A1: ( dom f) = ( Seg ( len f)) by FINSEQ_1:def 3;

      assume ( len f) <= n;

      hence thesis by A1, FINSEQ_1: 5, RELAT_1: 68;

    end;

    

     RFINSEQ6: for D be set, f be FinSequence of D, n,m be Nat holds n in ( dom f) & m in ( Seg n) implies ((f | n) . m) = (f . m) & m in ( dom f)

    proof

      let D be set, f be FinSequence of D, n,m be Nat;

      assume that

       A1: n in ( dom f) and

       A2: m in ( Seg n);

      

       A3: ( dom f) = ( Seg ( len f)) & n <= ( len f) by A1, FINSEQ_1:def 3, FINSEQ_3: 25;

      then

       A4: ( Seg n) c= ( dom f) by FINSEQ_1: 5;

      ( Seg n) = (( dom f) /\ ( Seg n)) by A3, FINSEQ_1: 5, XBOOLE_1: 28

      .= ( dom (f | n)) by RELAT_1: 61;

      hence thesis by A2, A4, FUNCT_1: 47;

    end;

    

     RFINSEQ8: for D be set, f be FinSequence of D, n be Nat holds ((f | n) ^ (f /^ n)) = f

    proof

      let D be set, f be FinSequence of D, n be Nat;

      set fn = (f /^ n);

      per cases ;

        suppose ( len f) < n;

        then (f /^ n) = ( <*> D) & (f | n) = f by RFINSEQ:def 1, RFINSEQlm3;

        hence thesis by FINSEQ_1: 34;

      end;

        suppose

         A1: n <= ( len f);

        then

         A3: ( len (f | n)) = n by FINSEQ_1: 59;

        

         A4: ( len fn) = (( len f) - n) by A1, RFINSEQ:def 1;

        then

         A5: ( len ((f | n) ^ (f /^ n))) = (n + (( len f) - n)) by A3, FINSEQ_1: 22;

        

         A6: ( dom (f | n)) = ( Seg n) by A3, FINSEQ_1:def 3;

        now

          let m be Nat;

          assume m in ( dom f);

          then

           A8: 1 <= m & m <= ( len f) by FINSEQ_3: 25;

          per cases ;

            suppose

             A10: m <= n;

            then 1 <= n by A8, XXREAL_0: 2;

            then

             A11: n in ( dom f) by A1, FINSEQ_3: 25;

            

             A12: m in ( Seg n) by A8, A10;

            

            hence (((f | n) ^ (f /^ n)) . m) = ((f | n) . m) by A6, FINSEQ_1:def 7

            .= (f . m) by A12, A11, RFINSEQ6;

          end;

            suppose

             A13: n < m;

            then ( max ( 0 ,(m - n))) = (m - n) by FINSEQ_2: 4;

            then

            reconsider k = (m - n) as Element of NAT by FINSEQ_2: 5;

            (n + 1) <= m by A13, NAT_1: 13;

            then 1 <= k by XREAL_1: 19;

            then

             A15: k in ( dom fn) by A4, A8, XREAL_1: 9, FINSEQ_3: 25;

            (((f | n) ^ (f /^ n)) . m) = (fn . k) by A3, A5, A8, A13, FINSEQ_1: 24;

            then (((f | n) ^ (f /^ n)) . m) = (f . (k + n)) by A1, A15, RFINSEQ:def 1;

            hence (((f | n) ^ (f /^ n)) . m) = (f . m);

          end;

        end;

        hence thesis by A5, FINSEQ_2: 9;

      end;

    end;

    theorem :: MEASURE9:6

    

     Th4: for D be set, Y be FinSequenceSet of D, F1,F2 be FinSequence of Y holds ( Length (F1 ^ F2)) = (( Length F1) ^ ( Length F2))

    proof

      let D be set, Y be FinSequenceSet of D, F1,F2 be FinSequence of Y;

      

       B1: ( dom ( Length (F1 ^ F2))) = ( dom (F1 ^ F2)) & ( dom ( Length F1)) = ( dom F1) & ( dom ( Length F2)) = ( dom F2) by Def1;

      then

       A1: ( len ( Length (F1 ^ F2))) = ( len (F1 ^ F2)) & ( len ( Length F1)) = ( len F1) & ( len ( Length F2)) = ( len F2) by FINSEQ_3: 29;

      

       B2: ( len (( Length F1) ^ ( Length F2))) = (( len ( Length F1)) + ( len ( Length F2))) by FINSEQ_1: 22;

      then

       A2: ( len ( Length (F1 ^ F2))) = ( len (( Length F1) ^ ( Length F2))) by A1, FINSEQ_1: 22;

      now

        let k be Nat;

        assume

         A3: 1 <= k & k <= ( len ( Length (F1 ^ F2)));

        then k in ( dom ( Length (F1 ^ F2))) by FINSEQ_3: 25;

        then

         A4: (( Length (F1 ^ F2)) . k) = ( len ((F1 ^ F2) . k)) by Def1;

        per cases ;

          suppose

           B5: k <= ( len ( Length F1));

          then

           A5: k in ( dom F1) & k in ( dom ( Length F1)) by B1, A3, FINSEQ_3: 25;

          

          then ((( Length F1) ^ ( Length F2)) . k) = (( Length F1) . k) by FINSEQ_1:def 7

          .= ( len (F1 . k)) by B5, Def1, B1, A3, FINSEQ_3: 25;

          hence (( Length (F1 ^ F2)) . k) = ((( Length F1) ^ ( Length F2)) . k) by A5, A4, FINSEQ_1:def 7;

        end;

          suppose

           A7: ( len ( Length F1)) < k;

          then (( len ( Length F1)) + 1) <= k by NAT_1: 13;

          then (k - (( len ( Length F1)) + 1)) is Nat by NAT_1: 21;

          then

          reconsider k1 = ((k - ( len ( Length F1))) - 1) as Nat;

          k <= (( len ( Length F1)) + ( len ( Length F2))) by A3, A1, FINSEQ_1: 22;

          then (k - ( len ( Length F1))) <= ( len ( Length F2)) by XREAL_1: 20;

          then

           A10: (k1 + 1) in ( dom ( Length F2)) by FINSEQ_3: 25, NAT_1: 11;

          ((( Length F1) ^ ( Length F2)) . k) = (( Length F2) . (k1 + 1)) by A2, A3, A7, FINSEQ_1: 24

          .= ( len (F2 . (k1 + 1))) by A10, Def1;

          hence (( Length (F1 ^ F2)) . k) = ((( Length F1) ^ ( Length F2)) . k) by A4, A3, A1, A7, FINSEQ_1: 24;

        end;

      end;

      hence thesis by B2, A1, FINSEQ_1: 22;

    end;

    theorem :: MEASURE9:7

    

     Th5: for D be set, Y be FinSequenceSet of D, F be FinSequence of Y, k1,k2 be Nat st k1 <= k2 holds ( Sum ( Length (F | k1))) <= ( Sum ( Length (F | k2)))

    proof

      let D be set, Y be FinSequenceSet of D, F be FinSequence of Y, k1,k2 be Nat;

      assume k1 <= k2;

      then ((F | k2) | k1) = (F | k1) by FINSEQ_1: 82;

      then (F | k2) = ((F | k1) ^ ((F | k2) /^ k1)) by RFINSEQ8;

      then ( Length (F | k2)) = (( Length (F | k1)) ^ ( Length ((F | k2) /^ k1))) by Th4;

      then ( Sum ( Length (F | k2))) = (( Sum ( Length (F | k1))) + ( Sum ( Length ((F | k2) /^ k1)))) by RVSUM_1: 75;

      hence thesis by NAT_1: 11;

    end;

    theorem :: MEASURE9:8

    

     Th6: for D be set, Y be FinSequenceSet of D, F be FinSequence of Y, m1,m2,k1,k2 be Nat st 1 <= m1 & 1 <= m2 & (m1 + ( Sum ( Length (F | k1)))) = (m2 + ( Sum ( Length (F | k2)))) & (m1 + ( Sum ( Length (F | k1)))) <= ( Sum ( Length (F | (k1 + 1)))) & (m2 + ( Sum ( Length (F | k2)))) <= ( Sum ( Length (F | (k2 + 1)))) holds m1 = m2 & k1 = k2

    proof

      let D be set, Y be FinSequenceSet of D, F be FinSequence of Y, m1,m2,k1,k2 be Nat;

      assume that

       A1: 1 <= m1 & 1 <= m2 and

       A2: (m1 + ( Sum ( Length (F | k1)))) = (m2 + ( Sum ( Length (F | k2)))) and

       A3: (m1 + ( Sum ( Length (F | k1)))) <= ( Sum ( Length (F | (k1 + 1)))) and

       A4: (m2 + ( Sum ( Length (F | k2)))) <= ( Sum ( Length (F | (k2 + 1))));

      set n = (m1 + ( Sum ( Length (F | k1))));

       A5:

      now

        assume

         A6: k1 <> k2;

        per cases by A6, XXREAL_0: 1;

          suppose k1 < k2;

          then (k1 + 1) <= k2 by NAT_1: 13;

          then ( Sum ( Length (F | (k1 + 1)))) <= ( Sum ( Length (F | k2))) by Th5;

          then n <= ( Sum ( Length (F | k2))) by A3, XXREAL_0: 2;

          hence contradiction by A2, A1, NAT_1: 19;

        end;

          suppose k1 > k2;

          then (k2 + 1) <= k1 by NAT_1: 13;

          then ( Sum ( Length (F | (k2 + 1)))) <= ( Sum ( Length (F | k1))) by Th5;

          then n <= ( Sum ( Length (F | k1))) by A2, A4, XXREAL_0: 2;

          hence contradiction by A1, NAT_1: 19;

        end;

      end;

      now

        assume m1 <> m2;

        then (( Sum ( Length (F | k1))) - ( Sum ( Length (F | k2)))) <> 0 by A2;

        hence k1 <> k2;

      end;

      hence thesis by A5;

    end;

    definition

      let D be non empty set, Y be FinSequenceSet of D, F be FinSequence of Y;

      :: MEASURE9:def2

      func joined_FinSeq F -> FinSequence of D means

      : Def2: ( len it ) = ( Sum ( Length F)) & for n be Nat st n in ( dom it ) holds ex k,m be Nat st 1 <= m & m <= ( len (F . (k + 1))) & k < ( len F) & (m + ( Sum ( Length (F | k)))) = n & n <= ( Sum ( Length (F | (k + 1)))) & (it . n) = ((F . (k + 1)) . m);

      existence

      proof

        defpred P[ Nat, object] means ex k,m be Nat st 1 <= m & m <= ( len (F . (k + 1))) & k < ( len F) & (m + ( Sum ( Length (F | k)))) = $1 & $1 <= ( Sum ( Length (F | (k + 1)))) & $2 = ((F . (k + 1)) . m);

        

         A1: for n be Nat st n in ( Seg ( Sum ( Length F))) holds ex x be Element of D st P[n, x]

        proof

          let n be Nat;

          assume n in ( Seg ( Sum ( Length F)));

          then 1 <= n & n <= ( Sum ( Length F)) by FINSEQ_1: 1;

          then

          consider k,m be Nat such that

           A2: 1 <= m & m <= ( len (F . (k + 1))) & k < ( len F) & (m + ( Sum ( Length (F | k)))) = n & n <= ( Sum ( Length (F | (k + 1)))) by Th3;

          m in ( dom (F . (k + 1))) by A2, FINSEQ_3: 25;

          then ((F . (k + 1)) . m) in ( rng (F . (k + 1))) by FUNCT_1: 3;

          then

          reconsider x = ((F . (k + 1)) . m) as Element of D;

          take x;

          thus thesis by A2;

        end;

        consider IT be FinSequence of D such that

         A3: ( dom IT) = ( Seg ( Sum ( Length F))) & for n be Nat st n in ( Seg ( Sum ( Length F))) holds P[n, (IT . n)] from FINSEQ_1:sch 5( A1);

        take IT;

        thus ( len IT) = ( Sum ( Length F)) by A3, FINSEQ_1:def 3;

        thus for n be Nat st n in ( dom IT) holds ex k,m be Nat st 1 <= m & m <= ( len (F . (k + 1))) & k < ( len F) & (m + ( Sum ( Length (F | k)))) = n & n <= ( Sum ( Length (F | (k + 1)))) & (IT . n) = ((F . (k + 1)) . m) by A3;

      end;

      uniqueness

      proof

        let IT1,IT2 be FinSequence of D;

        assume that

         A1: ( len IT1) = ( Sum ( Length F)) & (for n be Nat st n in ( dom IT1) holds ex k,m be Nat st 1 <= m & m <= ( len (F . (k + 1))) & k < ( len F) & (m + ( Sum ( Length (F | k)))) = n & n <= ( Sum ( Length (F | (k + 1)))) & (IT1 . n) = ((F . (k + 1)) . m)) and

         A2: ( len IT2) = ( Sum ( Length F)) & (for n be Nat st n in ( dom IT2) holds ex k,m be Nat st 1 <= m & m <= ( len (F . (k + 1))) & k < ( len F) & (m + ( Sum ( Length (F | k)))) = n & n <= ( Sum ( Length (F | (k + 1)))) & (IT2 . n) = ((F . (k + 1)) . m));

        

         A3: ( dom IT1) = ( dom IT2) by A1, A2, FINSEQ_3: 29;

        now

          let n be Nat;

          assume

           A4: n in ( dom IT1);

          then

          consider k1,m1 be Nat such that

           A5: 1 <= m1 & m1 <= ( len (F . (k1 + 1))) & k1 < ( len F) & (m1 + ( Sum ( Length (F | k1)))) = n & n <= ( Sum ( Length (F | (k1 + 1)))) & (IT1 . n) = ((F . (k1 + 1)) . m1) by A1;

          consider k2,m2 be Nat such that

           A6: 1 <= m2 & m2 <= ( len (F . (k2 + 1))) & k2 < ( len F) & (m2 + ( Sum ( Length (F | k2)))) = n & n <= ( Sum ( Length (F | (k2 + 1)))) & (IT2 . n) = ((F . (k2 + 1)) . m2) by A2, A3, A4;

          k1 = k2 & m1 = m2 by A5, A6, Th6;

          hence (IT1 . n) = (IT2 . n) by A5, A6;

        end;

        hence IT1 = IT2 by A1, A2, FINSEQ_3: 29, FINSEQ_1: 13;

      end;

    end

    definition

      let D be set, Y be FinSequenceSet of D, s be sequence of Y;

      :: MEASURE9:def3

      func Length s -> sequence of NAT means

      : Def3: for n be Nat holds (it . n) = ( len (s . n));

      existence

      proof

        defpred P[ Nat, object] means $2 = ( len (s . $1));

        

         A1: for k be Element of NAT holds ex x be Element of NAT st P[k, x];

        consider IT be Function of NAT , NAT such that

         A2: for k be Element of NAT holds P[k, (IT . k)] from FUNCT_2:sch 3( A1);

        take IT;

        hereby

          let n be Nat;

          n is Element of NAT by ORDINAL1:def 12;

          hence (IT . n) = ( len (s . n)) by A2;

        end;

      end;

      uniqueness

      proof

        let IT1,IT2 be sequence of NAT ;

        assume that

         A1: for n be Nat holds (IT1 . n) = ( len (s . n)) and

         A2: for n be Nat holds (IT2 . n) = ( len (s . n));

        now

          let n be Element of NAT ;

          (IT1 . n) = ( len (s . n)) by A1;

          hence (IT1 . n) = (IT2 . n) by A2;

        end;

        hence IT1 = IT2 by FUNCT_2: 63;

      end;

    end

    definition

      let s be sequence of NAT ;

      :: original: Partial_Sums

      redefine

      func Partial_Sums s -> sequence of NAT ;

      correctness

      proof

        

         A2: ( Partial_Sums s) is total;

        now

          let y be object;

          assume y in ( rng ( Partial_Sums s));

          then

          consider n be object such that

           A3: n in ( dom ( Partial_Sums s)) & y = (( Partial_Sums s) . n) by FUNCT_1:def 3;

          reconsider n as Nat by A3;

          defpred P[ Nat] means (( Partial_Sums s) . $1) is Nat;

          (( Partial_Sums s) . 0 ) = (s . 0 ) by SERIES_1:def 1;

          then

           A4: P[ 0 ];

          

           A5: for k be Nat st P[k] holds P[(k + 1)]

          proof

            let k be Nat;

            assume P[k];

            then

            reconsider Pk = (( Partial_Sums s) . k) as Nat;

            (( Partial_Sums s) . (k + 1)) = (Pk + (s . (k + 1))) by SERIES_1:def 1;

            hence P[(k + 1)];

          end;

          for k be Nat holds P[k] from NAT_1:sch 2( A4, A5);

          then (( Partial_Sums s) . n) is Nat;

          hence y in NAT by A3, ORDINAL1:def 12;

        end;

        hence ( Partial_Sums s) is sequence of NAT by A2, TARSKI:def 3, FUNCT_2: 2;

      end;

    end

    registration

      let D be non empty set;

      cluster non empty with_non-empty_element for FinSequenceSet of D;

      existence

      proof

        consider x be object such that

         A1: x in D by XBOOLE_0:def 1;

        reconsider x as Element of D by A1;

        set S = { <*x*>};

        for a be object st a in S holds a is FinSequence of D by TARSKI:def 1;

        then

        reconsider S as FinSequenceSet of D by FINSEQ_2:def 3;

        take S;

        thus S is non empty with_non-empty_element;

      end;

    end

    theorem :: MEASURE9:9

    

     Th7: for D be non empty set, Y be non empty with_non-empty_element FinSequenceSet of D, s be non-empty sequence of Y, n be Nat holds ( len (s . n)) >= 1 & n < (( Partial_Sums ( Length s)) . n) & (( Partial_Sums ( Length s)) . n) < (( Partial_Sums ( Length s)) . (n + 1))

    proof

      let D be non empty set, Y be non empty with_non-empty_element FinSequenceSet of D, s be non-empty sequence of Y, n be Nat;

      defpred P[ Nat] means $1 < (( Partial_Sums ( Length s)) . $1);

      

       A1: for k be Nat holds ( len (s . k)) >= 1

      proof

        let k be Nat;

        ( dom s) = NAT by FUNCT_2:def 1;

        then k in ( dom s) by ORDINAL1:def 12;

        hence ( len (s . k)) >= 1 by FINSEQ_1: 20;

      end;

      (( Partial_Sums ( Length s)) . 0 ) = (( Length s) . 0 ) by SERIES_1:def 1

      .= ( len (s . 0 )) by Def3;

      then

       A3: P[ 0 ];

      

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

      proof

        let k be Nat;

        assume

         A5: P[k];

        

         A6: (( Partial_Sums ( Length s)) . (k + 1)) = ((( Partial_Sums ( Length s)) . k) + (( Length s) . (k + 1))) by SERIES_1:def 1;

        (( Length s) . (k + 1)) = ( len (s . (k + 1))) by Def3;

        hence P[(k + 1)] by A1, A6, A5, XREAL_1: 8;

      end;

      for k be Nat holds P[k] from NAT_1:sch 2( A3, A4);

      hence ( len (s . n)) >= 1 & n < (( Partial_Sums ( Length s)) . n) by A1;

      (( Partial_Sums ( Length s)) . (n + 1)) = ((( Partial_Sums ( Length s)) . n) + (( Length s) . (n + 1))) by SERIES_1:def 1

      .= ((( Partial_Sums ( Length s)) . n) + ( len (s . (n + 1)))) by Def3;

      hence thesis by XREAL_1: 29;

    end;

    theorem :: MEASURE9:10

    

     Th8: for D be non empty set, Y be non empty with_non-empty_element FinSequenceSet of D, s be non-empty sequence of Y, n be Nat holds ex k,m be Nat st m in ( dom (s . k)) & ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) = n

    proof

      let D be non empty set, Y be non empty with_non-empty_element FinSequenceSet of D, s be non-empty sequence of Y, n be Nat;

      per cases ;

        suppose

         A1: n < ( len (s . 0 ));

        set k = 0 ;

        set m = (n + 1);

        take k, m;

        

         A4: m <= ( len (s . k)) by A1, NAT_1: 13;

        (( Partial_Sums ( Length s)) . k) = (( Length s) . 0 ) by SERIES_1:def 1

        .= ( len (s . k)) by Def3;

        hence m in ( dom (s . k)) & ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) = n by NAT_1: 11, A4, FINSEQ_3: 25;

      end;

        suppose

         A5: ( len (s . 0 )) <= n;

        then (( Length s) . 0 ) <= n by Def3;

        then

         A6: (( Partial_Sums ( Length s)) . 0 ) <= n by SERIES_1:def 1;

        now

          assume

           A8: for k be Nat st k < n holds n < (( Partial_Sums ( Length s)) . k) or (( Partial_Sums ( Length s)) . (k + 1)) <= n;

          defpred P[ Nat] means $1 < n implies (( Partial_Sums ( Length s)) . ($1 + 1)) <= n;

          

           A9: P[ 0 ] by A6, A8;

          

           A12: for k be Nat st P[k] holds P[(k + 1)] by A8, NAT_1: 13;

          

           A13: for k be Nat holds P[k] from NAT_1:sch 2( A9, A12);

          reconsider n1 = (n - 1) as Nat by A5, NAT_1: 20;

          (( Partial_Sums ( Length s)) . (n1 + 1)) <= n by A13, NAT_1: 19;

          hence contradiction by Th7;

        end;

        then

        consider k1 be Nat such that

         A14: k1 < n & (( Partial_Sums ( Length s)) . k1) <= n & n < (( Partial_Sums ( Length s)) . (k1 + 1));

        take k = (k1 + 1);

        reconsider m1 = ((( Partial_Sums ( Length s)) . k) - n) as Nat by A14, NAT_1: 21;

        (( Partial_Sums ( Length s)) . k) = ((( Partial_Sums ( Length s)) . k1) + (( Length s) . k)) by SERIES_1:def 1;

        then

         A15: m1 = (((( Partial_Sums ( Length s)) . k1) + ( len (s . k))) - n) by Def3;

        ((( Partial_Sums ( Length s)) . k1) - n) <= 0 by A14, XREAL_1: 47;

        then

         A17: (((( Partial_Sums ( Length s)) . k1) - n) + ( len (s . k))) <= ( len (s . k)) by XREAL_1: 32;

        then m1 <= ( len (s . k)) & ( len (s . k)) <= (( len (s . k)) + 1) by A15, NAT_1: 11;

        then

        reconsider m = ((( len (s . k)) + 1) - m1) as Nat by NAT_1: 21, XXREAL_0: 2;

        take m;

        m1 > 0 by A14, XREAL_1: 50;

        then (( len (s . k)) - m1) >= 0 & (1 - m1) <= 0 by A15, A17, NAT_1: 14, XREAL_1: 47, XREAL_1: 48;

        then ((( len (s . k)) - m1) + 1) >= ( 0 + 1) & (( len (s . k)) + (1 - m1)) <= (( len (s . k)) + 0 ) by XREAL_1: 6;

        hence m in ( dom (s . k)) & ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) = n by FINSEQ_3: 25;

      end;

    end;

    theorem :: MEASURE9:11

    

     Th9: for D be non empty set, Y be non empty with_non-empty_element FinSequenceSet of D, s be non-empty sequence of Y holds ( Partial_Sums ( Length s)) is increasing

    proof

      let D be non empty set, Y be non empty with_non-empty_element FinSequenceSet of D, s be non-empty sequence of Y;

      now

        let n,m be Nat;

        assume

         A1: n in ( dom ( Partial_Sums ( Length s))) & m in ( dom ( Partial_Sums ( Length s))) & n < m;

        defpred P[ Nat] means (( Partial_Sums ( Length s)) . n) < (( Partial_Sums ( Length s)) . ((n + 1) + $1));

        (( Partial_Sums ( Length s)) . ((n + 1) + 0 )) = ((( Partial_Sums ( Length s)) . n) + (( Length s) . (n + 1))) by SERIES_1:def 1

        .= ((( Partial_Sums ( Length s)) . n) + ( len (s . (n + 1)))) by Def3;

        then

         A3: P[ 0 ] by XREAL_1: 29;

        

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

        proof

          let k be Nat;

          assume

           A5: P[k];

          (( Partial_Sums ( Length s)) . ((n + 1) + (k + 1))) = ((( Partial_Sums ( Length s)) . ((n + 1) + k)) + (( Length s) . (((n + 1) + k) + 1))) by SERIES_1:def 1

          .= ((( Partial_Sums ( Length s)) . ((n + 1) + k)) + ( len (s . (((n + 1) + k) + 1)))) by Def3;

          then (( Partial_Sums ( Length s)) . ((n + 1) + (k + 1))) > (( Partial_Sums ( Length s)) . ((n + 1) + k)) by XREAL_1: 29;

          hence P[(k + 1)] by A5, XXREAL_0: 2;

        end;

        

         A7: for k be Nat holds P[k] from NAT_1:sch 2( A3, A4);

        (n + 1) <= m by A1, NAT_1: 13;

        then

        reconsider k = (m - (n + 1)) as Nat by NAT_1: 21;

        m = ((n + 1) + k);

        hence (( Partial_Sums ( Length s)) . n) < (( Partial_Sums ( Length s)) . m) by A7;

      end;

      hence ( Partial_Sums ( Length s)) is increasing by SEQM_3:def 1;

    end;

    theorem :: MEASURE9:12

    

     Th10: for D be non empty set, Y be non empty with_non-empty_element FinSequenceSet of D, s be non-empty sequence of Y, m1,m2,k1,k2 be Nat st m1 in ( dom (s . k1)) & m2 in ( dom (s . k2)) & (((( Partial_Sums ( Length s)) . k1) - ( len (s . k1))) + m1) = (((( Partial_Sums ( Length s)) . k2) - ( len (s . k2))) + m2) holds m1 = m2 & k1 = k2

    proof

      let D be non empty set, Y be non empty with_non-empty_element FinSequenceSet of D, s be non-empty sequence of Y, m1,m2,k1,k2 be Nat;

      assume that

       A1: m1 in ( dom (s . k1)) & m2 in ( dom (s . k2)) and

       A2: (((( Partial_Sums ( Length s)) . k1) - ( len (s . k1))) + m1) = (((( Partial_Sums ( Length s)) . k2) - ( len (s . k2))) + m2);

      set n = (((( Partial_Sums ( Length s)) . k1) - ( len (s . k1))) + m1);

      

       A3: 1 <= m1 & m1 <= ( len (s . k1)) & 1 <= m2 & m2 <= ( len (s . k2)) by A1, FINSEQ_3: 25;

      then (( len (s . k1)) - m1) >= 0 & (( len (s . k2)) - m2) >= 0 by XREAL_1: 48;

      then

       A4: ((( Partial_Sums ( Length s)) . k1) - (( len (s . k1)) - m1)) <= (( Partial_Sums ( Length s)) . k1) & ((( Partial_Sums ( Length s)) . k2) - (( len (s . k2)) - m2)) <= (( Partial_Sums ( Length s)) . k2) by XREAL_1: 43;

      

       A5: ( dom ( Partial_Sums ( Length s))) = NAT by FUNCT_2:def 1;

      then

       A6: k1 in ( dom ( Partial_Sums ( Length s))) & k2 in ( dom ( Partial_Sums ( Length s))) by ORDINAL1:def 12;

      

       A7: ( Partial_Sums ( Length s)) is increasing by Th9;

       A14:

      now

        assume

         A8: k1 <> k2;

        per cases by A8, XXREAL_0: 1;

          suppose k1 < k2;

          then

           A10: (k1 + 1) <= k2 by NAT_1: 13;

          1 <= (k1 + 1) by NAT_1: 11;

          then

          reconsider j = (k2 - 1) as Element of NAT by NAT_1: 21, A10, XXREAL_0: 2;

          

           A11: k1 <= j by A10, XREAL_1: 19;

          

           A12: (( Partial_Sums ( Length s)) . k1) <= (( Partial_Sums ( Length s)) . j)

          proof

            k1 = j or k1 < j by A11, XXREAL_0: 1;

            hence thesis by A5, A6, A7, SEQM_3:def 1;

          end;

          (( Partial_Sums ( Length s)) . (j + 1)) = ((( Partial_Sums ( Length s)) . j) + (( Length s) . (j + 1))) by SERIES_1:def 1

          .= ((( Partial_Sums ( Length s)) . j) + ( len (s . k2))) by Def3;

          then n > (( Partial_Sums ( Length s)) . j) by A3, A2, XREAL_1: 29;

          hence contradiction by A4, A12, XXREAL_0: 2;

        end;

          suppose k2 < k1;

          then

           A10: (k2 + 1) <= k1 by NAT_1: 13;

          1 <= (k2 + 1) by NAT_1: 11;

          then

          reconsider j = (k1 - 1) as Element of NAT by NAT_1: 21, A10, XXREAL_0: 2;

          

           A11: k2 <= j by A10, XREAL_1: 19;

          

           A12: (( Partial_Sums ( Length s)) . k2) <= (( Partial_Sums ( Length s)) . j)

          proof

            k2 = j or k2 < j by A11, XXREAL_0: 1;

            hence thesis by A5, A6, A7, SEQM_3:def 1;

          end;

          (( Partial_Sums ( Length s)) . (j + 1)) = ((( Partial_Sums ( Length s)) . j) + (( Length s) . (j + 1))) by SERIES_1:def 1

          .= ((( Partial_Sums ( Length s)) . j) + ( len (s . k1))) by Def3;

          then n > (( Partial_Sums ( Length s)) . j) by A3, XREAL_1: 29;

          hence contradiction by A2, A4, A12, XXREAL_0: 2;

        end;

      end;

      then ((( Partial_Sums ( Length s)) . k1) - ( len (s . k1))) = ((( Partial_Sums ( Length s)) . k2) - ( len (s . k2)));

      hence m1 = m2 & k1 = k2 by A2, A14;

    end;

    theorem :: MEASURE9:13

    

     Th11: for D be non empty set, Y be with_non-empty_element FinSequenceSet of D, s be non-empty sequence of Y holds ex N be increasing sequence of NAT st for k be Nat holds (N . k) = ((( Partial_Sums ( Length s)) . k) - 1)

    proof

      let D be non empty set, Y be with_non-empty_element FinSequenceSet of D, s be non-empty sequence of Y;

      defpred P[ Nat, Nat] means $2 = ((( Partial_Sums ( Length s)) . $1) - 1);

      

       A1: for k be Element of NAT holds ex n be Element of NAT st P[k, n]

      proof

        let k be Element of NAT ;

        reconsider n = ((( Partial_Sums ( Length s)) . k) - 1) as Element of NAT by Th7, NAT_1: 20;

        take n;

        thus thesis;

      end;

      consider N be Function of NAT , NAT such that

       A2: for k be Element of NAT holds P[k, (N . k)] from FUNCT_2:sch 3( A1);

      

       A3: for k be Nat holds (N . k) = ((( Partial_Sums ( Length s)) . k) - 1)

      proof

        let k be Nat;

        k is Element of NAT by ORDINAL1:def 12;

        hence thesis by A2;

      end;

      for n be Nat holds (N . n) < (N . (n + 1))

      proof

        let n be Nat;

        ((( Partial_Sums ( Length s)) . n) - 1) < ((( Partial_Sums ( Length s)) . (n + 1)) - 1) by Th7, XREAL_1: 9;

        then (N . n) < ((( Partial_Sums ( Length s)) . (n + 1)) - 1) by A3;

        hence (N . n) < (N . (n + 1)) by A3;

      end;

      then

      reconsider N as increasing sequence of NAT by VALUED_1:def 13;

      take N;

      thus thesis by A3;

    end;

    definition

      let D be non empty set, Y be with_non-empty_element FinSequenceSet of D, s be non-empty sequence of Y;

      :: MEASURE9:def4

      func joined_seq s -> sequence of D means

      : Def4: for n be Nat holds ex k,m be Nat st m in ( dom (s . k)) & ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) = n & (it . n) = ((s . k) . m);

      existence

      proof

        defpred P[ Nat, object] means ex k,m be Nat st m in ( dom (s . k)) & ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) = $1 & $2 = ((s . k) . m);

        

         A1: for n be Element of NAT holds ex y be Element of D st P[n, y]

        proof

          let n be Element of NAT ;

          consider k,m be Nat such that

           A2: m in ( dom (s . k)) & ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) = n by Th8;

          ((s . k) . m) in ( rng (s . k)) by A2, FUNCT_1: 3;

          then

          reconsider y = ((s . k) . m) as Element of D;

          take y;

          thus thesis by A2;

        end;

        consider IT be Function of NAT , D such that

         A4: for n be Element of NAT holds P[n, (IT . n)] from FUNCT_2:sch 3( A1);

        reconsider IT as sequence of D;

        take IT;

        hereby

          let n be Nat;

          n is Element of NAT by ORDINAL1:def 12;

          hence ex k,m be Nat st m in ( dom (s . k)) & ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) = n & (IT . n) = ((s . k) . m) by A4;

        end;

      end;

      uniqueness

      proof

        let f1,f2 be sequence of D such that

         A1: (for n be Nat holds ex k,m be Nat st m in ( dom (s . k)) & ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) = n & (f1 . n) = ((s . k) . m)) and

         A2: (for n be Nat holds ex k,m be Nat st m in ( dom (s . k)) & ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) = n & (f2 . n) = ((s . k) . m));

        for n be Element of NAT holds (f1 . n) = (f2 . n)

        proof

          let n be Element of NAT ;

          consider k1,m1 be Nat such that

           A3: m1 in ( dom (s . k1)) & ((((( Partial_Sums ( Length s)) . k1) - ( len (s . k1))) + m1) - 1) = n & (f1 . n) = ((s . k1) . m1) by A1;

          consider k2,m2 be Nat such that

           A4: m2 in ( dom (s . k2)) & ((((( Partial_Sums ( Length s)) . k2) - ( len (s . k2))) + m2) - 1) = n & (f2 . n) = ((s . k2) . m2) by A2;

          m1 = m2 & k1 = k2 by A3, A4, Th10;

          hence (f1 . n) = (f2 . n) by A3, A4;

        end;

        hence f1 = f2 by FUNCT_2:def 8;

      end;

    end

    theorem :: MEASURE9:14

    for D be non empty set, Y be with_non-empty_element FinSequenceSet of D, s be non-empty sequence of Y, s1 be sequence of D st (for n be Nat holds (s1 . n) = (( joined_seq s) . ((( Partial_Sums ( Length s)) . n) - 1))) holds s1 is subsequence of ( joined_seq s)

    proof

      let D be non empty set, Y be with_non-empty_element FinSequenceSet of D;

      let s be non-empty sequence of Y, s1 be sequence of D;

      assume

       A1: for n be Nat holds (s1 . n) = (( joined_seq s) . ((( Partial_Sums ( Length s)) . n) - 1));

      consider N be increasing sequence of NAT such that

       A2: for n be Nat holds (N . n) = ((( Partial_Sums ( Length s)) . n) - 1) by Th11;

      for n be Element of NAT holds (s1 . n) = ((( joined_seq s) * N) . n)

      proof

        let n be Element of NAT ;

        (s1 . n) = (( joined_seq s) . ((( Partial_Sums ( Length s)) . n) - 1)) by A1;

        then (s1 . n) = (( joined_seq s) . (N . n)) by A2;

        hence (s1 . n) = ((( joined_seq s) * N) . n) by FUNCT_2: 15;

      end;

      hence s1 is subsequence of ( joined_seq s) by FUNCT_2:def 8;

    end;

    theorem :: MEASURE9:15

    

     Th13: for D be non empty set, Y be with_non-empty_element FinSequenceSet of D, s be non-empty sequence of Y, k,m be Nat st m in ( dom (s . k)) holds ex n be Nat st n = ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) & (( joined_seq s) . n) = ((s . k) . m)

    proof

      let D be non empty set, Y be with_non-empty_element FinSequenceSet of D, s be non-empty sequence of Y, k,m be Nat;

      assume

       A0: m in ( dom (s . k));

      then

       A1: 1 <= m & m <= ( len (s . k)) by FINSEQ_3: 25;

      now

        per cases ;

          suppose

           A2: k = 0 ;

          

          then (( Partial_Sums ( Length s)) . k) = (( Length s) . 0 ) by SERIES_1:def 1

          .= ( len (s . 0 )) by Def3;

          hence ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) is Nat by A1, A2, NAT_1: 21;

        end;

          suppose k <> 0 ;

          then

          reconsider k1 = (k - 1) as Element of NAT by NAT_1: 14, NAT_1: 21;

          k = (k1 + 1);

          

          then (( Partial_Sums ( Length s)) . k) = ((( Partial_Sums ( Length s)) . k1) + (( Length s) . k)) by SERIES_1:def 1

          .= ((( Partial_Sums ( Length s)) . k1) + ( len (s . k))) by Def3;

          then

          reconsider n1 = ((( Partial_Sums ( Length s)) . k) - ( len (s . k))) as Nat;

          (n1 + m) >= m by NAT_1: 11;

          hence ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) is Nat by A1, XXREAL_0: 2, NAT_1: 21;

        end;

      end;

      then

      reconsider n = ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) as Nat;

      take n;

      thus n = ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1);

      consider k2,m2 be Nat such that

       A4: m2 in ( dom (s . k2)) & ((((( Partial_Sums ( Length s)) . k2) - ( len (s . k2))) + m2) - 1) = n & (( joined_seq s) . n) = ((s . k2) . m2) by Def4;

      m = m2 & k = k2 by A0, A4, Th10;

      hence (( joined_seq s) . n) = ((s . k) . m) by A4;

    end;

    theorem :: MEASURE9:16

    

     Th14: for D be non empty set, Y be FinSequenceSet of D, F be FinSequence of Y st (for n,m be Nat st n <> m holds ( union ( rng (F . n))) misses ( union ( rng (F . m)))) & (for n be Nat holds (F . n) is disjoint_valued) holds ( joined_FinSeq F) is disjoint_valued

    proof

      let D be non empty set, Y be FinSequenceSet of D, F be FinSequence of Y;

      assume that

       A1: for n,m be Nat st n <> m holds ( union ( rng (F . n))) misses ( union ( rng (F . m))) and

       A2: for n be Nat holds (F . n) is disjoint_valued;

      now

        let x,y be object;

        assume

         A3: x <> y;

        per cases ;

          suppose

           A4: x in ( dom ( joined_FinSeq F)) & y in ( dom ( joined_FinSeq F));

          then

          reconsider n1 = x, n2 = y as Nat;

          consider k1,m1 be Nat such that

           A5: 1 <= m1 & m1 <= ( len (F . (k1 + 1))) & k1 < ( len F) & (m1 + ( Sum ( Length (F | k1)))) = n1 & n1 <= ( Sum ( Length (F | (k1 + 1)))) & (( joined_FinSeq F) . x) = ((F . (k1 + 1)) . m1) by A4, Def2;

          consider k2,m2 be Nat such that

           A6: 1 <= m2 & m2 <= ( len (F . (k2 + 1))) & k2 < ( len F) & (m2 + ( Sum ( Length (F | k2)))) = n2 & n2 <= ( Sum ( Length (F | (k2 + 1)))) & (( joined_FinSeq F) . y) = ((F . (k2 + 1)) . m2) by A4, Def2;

          m1 in ( dom (F . (k1 + 1))) & m2 in ( dom (F . (k2 + 1))) by A5, A6, FINSEQ_3: 25;

          then

           A8: (( joined_FinSeq F) . x) in ( rng (F . (k1 + 1))) & (( joined_FinSeq F) . y) in ( rng (F . (k2 + 1))) by A5, A6, FUNCT_1: 3;

          now

            assume

             A9: not (( joined_FinSeq F) . x) misses (( joined_FinSeq F) . y);

            then ((( joined_FinSeq F) . x) /\ (( joined_FinSeq F) . y)) <> {} by XBOOLE_0:def 7;

            then

            consider z be object such that

             A10: z in ((( joined_FinSeq F) . x) /\ (( joined_FinSeq F) . y)) by XBOOLE_0:def 1;

            z in (( joined_FinSeq F) . x) & z in (( joined_FinSeq F) . y) by A10, XBOOLE_0:def 4;

            then z in ( union ( rng (F . (k1 + 1)))) & z in ( union ( rng (F . (k2 + 1)))) by A8, TARSKI:def 4;

            then

             A11: (k1 + 1) = (k2 + 1) by A1, XBOOLE_0: 3;

            (F . (k1 + 1)) is disjoint_valued by A2;

            hence contradiction by A5, A6, A3, A9, A11, PROB_2:def 2;

          end;

          hence (( joined_FinSeq F) . x) misses (( joined_FinSeq F) . y);

        end;

          suppose not x in ( dom ( joined_FinSeq F)) or not y in ( dom ( joined_FinSeq F));

          then (( joined_FinSeq F) . x) = {} or (( joined_FinSeq F) . y) = {} by FUNCT_1:def 2;

          hence (( joined_FinSeq F) . x) misses (( joined_FinSeq F) . y) by XBOOLE_1: 65;

        end;

      end;

      hence ( joined_FinSeq F) is disjoint_valued by PROB_2:def 2;

    end;

    theorem :: MEASURE9:17

    

     Th15: for D be non empty set, Y be FinSequenceSet of D, F be FinSequence of Y holds ( rng ( joined_FinSeq F)) = ( union { ( rng (F . n)) where n be Nat : n in ( dom F) })

    proof

      let D be non empty set, Y be FinSequenceSet of D, F be FinSequence of Y;

      now

        let x be object;

        assume x in ( rng ( joined_FinSeq F));

        then

        consider n be object such that

         A1: n in ( dom ( joined_FinSeq F)) & x = (( joined_FinSeq F) . n) by FUNCT_1:def 3;

        reconsider n as Nat by A1;

        consider k,m be Nat such that

         A2: 1 <= m & m <= ( len (F . (k + 1))) & k < ( len F) & (m + ( Sum ( Length (F | k)))) = n & n <= ( Sum ( Length (F | (k + 1)))) & (( joined_FinSeq F) . n) = ((F . (k + 1)) . m) by A1, Def2;

        1 <= (k + 1) & (k + 1) <= ( len F) by A2, NAT_1: 11, NAT_1: 13;

        then

         A3: (k + 1) in ( dom F) by FINSEQ_3: 25;

        m in ( dom (F . (k + 1))) by A2, FINSEQ_3: 25;

        then

         A4: x in ( rng (F . (k + 1))) by A1, A2, FUNCT_1: 3;

        ( rng (F . (k + 1))) in { ( rng (F . n)) where n be Nat : n in ( dom F) } by A3;

        hence x in ( union { ( rng (F . n)) where n be Nat : n in ( dom F) }) by A4, TARSKI:def 4;

      end;

      then

       A5: ( rng ( joined_FinSeq F)) c= ( union { ( rng (F . n)) where n be Nat : n in ( dom F) }) by TARSKI:def 3;

      now

        let x be object;

        assume x in ( union { ( rng (F . n)) where n be Nat : n in ( dom F) });

        then

        consider A be set such that

         A6: x in A & A in { ( rng (F . n)) where n be Nat : n in ( dom F) } by TARSKI:def 4;

        consider k be Nat such that

         A7: A = ( rng (F . k)) & k in ( dom F) by A6;

        consider m be object such that

         A8: m in ( dom (F . k)) & x = ((F . k) . m) by A6, A7, FUNCT_1:def 3;

        reconsider m as Nat by A8;

        

         A9: 1 <= k & k <= ( len F) by A7, FINSEQ_3: 25;

        reconsider k1 = (k - 1) as Nat by A7, FINSEQ_3: 25, NAT_1: 21;

        set n = (m + ( Sum ( Length (F | k1))));

        ( Length (F | (k1 + 1))) = (( Length (F | k1)) ^ <*( len (F . (k1 + 1)))*>) by Th2, A9, NAT_1: 13;

        then

         A11: ( Sum ( Length (F | (k1 + 1)))) = (( Sum ( Length (F | k1))) + ( len (F . (k1 + 1)))) by RVSUM_1: 74;

        

         A14: 1 <= m & m <= ( len (F . (k1 + 1))) by A8, FINSEQ_3: 25;

        then

         A12: n <= ( Sum ( Length (F | (k1 + 1)))) by A11, XREAL_1: 6;

        ( Sum ( Length (F | (k1 + 1)))) <= ( Sum ( Length (F | ( len F)))) by A9, Th5;

        then n <= ( Sum ( Length (F | ( len F)))) by A12, XXREAL_0: 2;

        then n <= ( Sum ( Length F)) by FINSEQ_1: 58;

        then

         A13: n <= ( len ( joined_FinSeq F)) by Def2;

        m <= n by NAT_1: 11;

        then 1 <= n by A14, XXREAL_0: 2;

        then

         A17: n in ( dom ( joined_FinSeq F)) by A13, FINSEQ_3: 25;

        then

        consider k2,m2 be Nat such that

         A15: 1 <= m2 & m2 <= ( len (F . (k2 + 1))) & k2 < ( len F) & (m2 + ( Sum ( Length (F | k2)))) = n & n <= ( Sum ( Length (F | (k2 + 1)))) & (( joined_FinSeq F) . n) = ((F . (k2 + 1)) . m2) by Def2;

        m = m2 & k1 = k2 by A14, A15, A12, Th6;

        hence x in ( rng ( joined_FinSeq F)) by A8, A15, A17, FUNCT_1: 3;

      end;

      then ( union { ( rng (F . n)) where n be Nat : n in ( dom F) }) c= ( rng ( joined_FinSeq F)) by TARSKI:def 3;

      hence thesis by A5, XBOOLE_0:def 10;

    end;

    begin

    definition

      let x be ext-real number;

      :: original: <*

      redefine

      func <*x*> -> FinSequence of ExtREAL ;

      coherence

      proof

        now

          let y be object;

          assume y in ( rng <*x*>);

          then y in {x} by FINSEQ_1: 39;

          hence y in ExtREAL by XXREAL_0:def 1;

        end;

        hence thesis by TARSKI:def 3, FINSEQ_1:def 4;

      end;

    end

    definition

      let e be FinSequence of ( ExtREAL * );

      :: MEASURE9:def5

      func Sum e -> FinSequence of ExtREAL means

      : Def5: ( len it ) = ( len e) & for k be Nat st k in ( dom it ) holds (it . k) = ( Sum (e . k));

      existence

      proof

        deffunc f( Nat) = ( Sum (e . $1));

        consider e1 be FinSequence of ExtREAL such that

         A1: ( len e1) = ( len e) & for k be Nat st k in ( dom e1) holds (e1 . k) = f(k) from FINSEQ_2:sch 1;

        take e1;

        thus thesis by A1;

      end;

      uniqueness

      proof

        let e1,e2 be FinSequence of ExtREAL such that

         A2: ( len e1) = ( len e) and

         A3: for k be Nat st k in ( dom e1) holds (e1 . k) = ( Sum (e . k)) and

         A4: ( len e2) = ( len e) and

         A5: for k be Nat st k in ( dom e2) holds (e2 . k) = ( Sum (e . k));

        ( dom e1) = ( dom e2) & for k be Nat st k in ( dom e1) holds (e1 . k) = (e2 . k)

        proof

          

          thus

           A6: ( dom e1) = ( Seg ( len e)) by A2, FINSEQ_1:def 3

          .= ( dom e2) by A4, FINSEQ_1:def 3;

          let k be Nat such that

           A7: k in ( dom e1);

          

          thus (e1 . k) = ( Sum (e . k)) by A3, A7

          .= (e2 . k) by A5, A6, A7;

        end;

        hence thesis by FINSEQ_1: 13;

      end;

    end

    definition

      let M be Matrix of ExtREAL ;

      :: MEASURE9:def6

      func SumAll M -> Element of ExtREAL equals ( Sum ( Sum M));

      coherence ;

    end

    theorem :: MEASURE9:18

    

     Th16: for M be Matrix of ExtREAL holds ( len ( Sum M)) = ( len M) & for i be Nat st i in ( Seg ( len M)) holds (( Sum M) . i) = ( Sum ( Line (M,i)))

    proof

      let M be Matrix of ExtREAL ;

      thus ( len ( Sum M)) = ( len M) by Def5;

      thus for k be Nat st k in ( Seg ( len M)) holds (( Sum M) . k) = ( Sum ( Line (M,k)))

      proof

        let k be Nat such that

         A1: k in ( Seg ( len M));

        

         A2: k in ( dom M) by A1, FINSEQ_1:def 3;

        k in ( Seg ( len ( Sum M))) by A1, Def5;

        then k in ( dom ( Sum M)) by FINSEQ_1:def 3;

        

        hence (( Sum M) . k) = ( Sum (M . k)) by Def5

        .= ( Sum ( Line (M,k))) by A2, MATRIX_0: 60;

      end;

    end;

    theorem :: MEASURE9:19

    

     Th17: for F be FinSequence of ExtREAL st for i be Nat st i in ( dom F) holds (F . i) <> -infty holds ( Sum F) <> -infty

    proof

      let F be FinSequence of ExtREAL ;

      assume

       A1: for i be Nat st i in ( dom F) holds (F . i) <> -infty ;

      consider f be Function of NAT , ExtREAL such that

       A2: ( Sum F) = (f . ( len F)) & (f . 0 ) = 0 & for i be Nat st i < ( len F) holds (f . (i + 1)) = ((f . i) + (F . (i + 1))) by EXTREAL1:def 2;

      defpred P[ Nat] means $1 <= ( len F) implies (f . $1) <> -infty ;

      

       A4: P[ 0 ] by A2;

      

       A5: for j be Nat st P[j] holds P[(j + 1)]

      proof

        let j be Nat;

        assume

         A6: P[j];

        now

          assume

           B2: (j + 1) <= ( len F);

          then

           A8: (f . (j + 1)) = ((f . j) + (F . (j + 1))) by A2, NAT_1: 13;

          1 <= (j + 1) by NAT_1: 11;

          then (F . (j + 1)) <> -infty by A1, B2, FINSEQ_3: 25;

          hence (f . (j + 1)) <> -infty by A8, A6, B2, NAT_1: 13, XXREAL_3: 17;

        end;

        hence P[(j + 1)];

      end;

      for i be Nat holds P[i] from NAT_1:sch 2( A4, A5);

      hence ( Sum F) <> -infty by A2;

    end;

    theorem :: MEASURE9:20

    

     Th18: for F,G,H be FinSequence of ExtREAL st not -infty in ( rng F) & not -infty in ( rng G) & ( dom F) = ( dom G) & H = (F + G) holds ( Sum H) = (( Sum F) + ( Sum G))

    proof

      let F,G,H be FinSequence of ExtREAL ;

      assume that

       A1: not -infty in ( rng F) & not -infty in ( rng G) and

       A3: ( dom F) = ( dom G) and

       A4: H = (F + G);

      

       B1: for y be object st y in ( rng F) holds not y in { -infty } by A1, TARSKI:def 1;

      then

       A7: (F " { -infty }) = {} by XBOOLE_0: 3, RELAT_1: 138;

      

       B2: for y be object st y in ( rng G) holds not y in { -infty } by A1, TARSKI:def 1;

      then

       A10: (G " { -infty }) = {} by XBOOLE_0: 3, RELAT_1: 138;

      

       A11: ( dom H) = ((( dom F) /\ ( dom G)) \ (((F " { -infty }) /\ (G " { +infty })) \/ ((F " { +infty }) /\ (G " { -infty })))) by A4, MESFUNC1:def 3

      .= ( dom F) by A3, A7, A10;

      then

       A12: ( len H) = ( len F) by FINSEQ_3: 29;

      consider h be Function of NAT , ExtREAL such that

       A13: ( Sum H) = (h . ( len H)) & (h . 0 ) = 0. & for i be Nat st i < ( len H) holds (h . (i + 1)) = ((h . i) + (H . (i + 1))) by EXTREAL1:def 2;

      consider f be Function of NAT , ExtREAL such that

       A16: ( Sum F) = (f . ( len F)) & (f . 0 ) = 0. & for i be Nat st i < ( len F) holds (f . (i + 1)) = ((f . i) + (F . (i + 1))) by EXTREAL1:def 2;

      consider g be Function of NAT , ExtREAL such that

       A19: ( Sum G) = (g . ( len G)) & (g . 0 ) = 0. & for i be Nat st i < ( len G) holds (g . (i + 1)) = ((g . i) + (G . (i + 1))) by EXTREAL1:def 2;

      defpred P[ Nat] means $1 <= ( len H) implies (h . $1) = ((f . $1) + (g . $1));

      

       A22: ( len H) = ( len G) by A3, A11, FINSEQ_3: 29;

      

       A23: for k be Nat st P[k] holds P[(k + 1)]

      proof

        let k be Nat;

        assume

         A24: P[k];

        assume

         A25: (k + 1) <= ( len H);

        

         A26: k < ( len H) by A25, NAT_1: 13;

        

         A27: (f . (k + 1)) = ((f . k) + (F . (k + 1))) & (g . (k + 1)) = ((g . k) + (G . (k + 1))) by A16, A19, A12, A22, A25, NAT_1: 13;

        

         A28: (k + 1) in ( dom H) by A25, NAT_1: 11, FINSEQ_3: 25;

        

         A29: (f . k) <> -infty & (g . k) <> -infty & (F . (k + 1)) <> -infty & (G . (k + 1)) <> -infty

        proof

          defpred Pg[ Nat] means $1 <= ( len H) implies (g . $1) <> -infty ;

          defpred Pf[ Nat] means $1 <= ( len H) implies (f . $1) <> -infty ;

          

           A30: for m be Nat st Pf[m] holds Pf[(m + 1)]

          proof

            let m be Nat;

            assume

             A31: Pf[m];

            assume

             A32: (m + 1) <= ( len H);

            then (m + 1) in ( dom H) by NAT_1: 11, FINSEQ_3: 25;

            then not (F . (m + 1)) in { -infty } by B1, A11, FUNCT_1: 3;

            then

             A33: (F . (m + 1)) <> -infty by TARSKI:def 1;

            (f . (m + 1)) = ((f . m) + (F . (m + 1))) by A12, A16, A32, NAT_1: 13;

            hence thesis by A33, A32, NAT_1: 13, A31, XXREAL_3: 17;

          end;

          

           A34: Pf[ 0 ] by A16;

          for i be Nat holds Pf[i] from NAT_1:sch 2( A34, A30);

          hence (f . k) <> -infty by A26;

          

           A35: for m be Nat st Pg[m] holds Pg[(m + 1)]

          proof

            let m be Nat;

            assume

             A36: Pg[m];

            assume

             A37: (m + 1) <= ( len H);

            then (m + 1) in ( dom H) by NAT_1: 11, FINSEQ_3: 25;

            then not (G . (m + 1)) in { -infty } by B2, A11, A3, FUNCT_1: 3;

            then

             A38: (G . (m + 1)) <> -infty by TARSKI:def 1;

            (g . (m + 1)) = ((g . m) + (G . (m + 1))) by A19, A22, A37, NAT_1: 13;

            hence thesis by A38, A37, NAT_1: 13, A36, XXREAL_3: 17;

          end;

          

           A39: Pg[ 0 ] by A19;

          for i be Nat holds Pg[i] from NAT_1:sch 2( A39, A35);

          hence (g . k) <> -infty by A26;

          thus (F . (k + 1)) <> -infty by A1, A11, A28, FUNCT_1: 3;

          thus thesis by A1, A3, A11, A28, FUNCT_1: 3;

        end;

        then

         A40: ((f . k) + (F . (k + 1))) <> -infty by XXREAL_3: 17;

        

         A41: (h . (k + 1)) = (((f . k) + (g . k)) + (H . (k + 1))) by A13, A24, A25, NAT_1: 13

        .= (((f . k) + (g . k)) + ((F . (k + 1)) + (G . (k + 1)))) by A4, A28, MESFUNC1:def 3;

        ((f . k) + (g . k)) <> -infty by A29, XXREAL_3: 17;

        

        then (h . (k + 1)) = ((((f . k) + (g . k)) + (F . (k + 1))) + (G . (k + 1))) by A41, A29, XXREAL_3: 29

        .= ((((f . k) + (F . (k + 1))) + (g . k)) + (G . (k + 1))) by A29, XXREAL_3: 29

        .= ((f . (k + 1)) + (g . (k + 1))) by A27, A29, A40, XXREAL_3: 29;

        hence thesis;

      end;

      

       A42: P[ 0 ] by A16, A19, A13;

      for i be Nat holds P[i] from NAT_1:sch 2( A42, A23);

      hence thesis by A16, A19, A13, A12, A22;

    end;

    theorem :: MEASURE9:21

    

     Th19: for r be R_eal, F be FinSequence of ExtREAL holds ( Sum (F ^ <*r*>)) = (( Sum F) + r)

    proof

      let r be R_eal, F be FinSequence of ExtREAL ;

      consider f be Function of NAT , ExtREAL such that

       A1: ( Sum (F ^ <*r*>)) = (f . ( len (F ^ <*r*>))) & (f . 0 ) = 0 & for i be Nat st i < ( len (F ^ <*r*>)) holds (f . (i + 1)) = ((f . i) + ((F ^ <*r*>) . (i + 1))) by EXTREAL1:def 2;

      consider g be Function of NAT , ExtREAL such that

       A2: ( Sum F) = (g . ( len F)) & (g . 0 ) = 0 & for i be Nat st i < ( len F) holds (g . (i + 1)) = ((g . i) + (F . (i + 1))) by EXTREAL1:def 2;

      ( len (F ^ <*r*>)) = (( len F) + ( len <*r*>)) by FINSEQ_1: 22;

      then

       B1: ( len (F ^ <*r*>)) = (( len F) + 1) by FINSEQ_1: 39;

      then

       B2: ( len F) < ( len (F ^ <*r*>)) by NAT_1: 13;

      defpred P[ Nat] means $1 <= ( len F) implies (f . $1) = (g . $1);

      

       A3: P[ 0 ] by A1, A2;

      

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

      proof

        let k be Nat;

        assume

         A5: P[k];

        assume

         A6: (k + 1) <= ( len F);

        then

         A7: k < ( len F) by NAT_1: 13;

        

         A9: ((F ^ <*r*>) . (k + 1)) = (F . (k + 1)) by A6, FINSEQ_1: 64, NAT_1: 11;

        k < ( len (F ^ <*r*>)) by A7, B1, NAT_1: 13;

        then (f . (k + 1)) = ((f . k) + ((F ^ <*r*>) . (k + 1))) by A1;

        hence (f . (k + 1)) = (g . (k + 1)) by A2, A6, A5, A9, NAT_1: 13;

      end;

      for i be Nat holds P[i] from NAT_1:sch 2( A3, A4);

      then (f . ( len F)) = (g . ( len F));

      then (f . (( len F) + 1)) = ((g . ( len F)) + ((F ^ <*r*>) . (( len F) + 1))) by A1, B2;

      hence ( Sum (F ^ <*r*>)) = (( Sum F) + r) by A1, A2, B1, FINSEQ_1: 42;

    end;

    theorem :: MEASURE9:22

    

     Th20: for r be R_eal, i be Nat st r is real holds ( Sum (i |-> r)) = (i * r)

    proof

      let r be R_eal, i be Nat;

      assume

       A0: r is real;

      defpred P[ Nat] means ( Sum ($1 |-> r)) = ($1 * r);

      

       A1: for i be Nat st P[i] holds P[(i + 1)]

      proof

        let i be Nat such that

         A2: ( Sum (i |-> r)) = (i * r);

        reconsider i1 = i, One = 1 as ext-real number;

        

        thus ( Sum ((i + 1) |-> r)) = ( Sum ((i |-> r) ^ <*r*>)) by FINSEQ_2: 60

        .= ((i * r) + r) by A2, Th19

        .= ((i * r) + (1 * r)) by XXREAL_3: 81

        .= ((i1 + One) * r) by A0, XXREAL_3: 95

        .= ((i + 1) * r) by XXREAL_3:def 2;

      end;

      

       A3: P[ 0 ] by EXTREAL1: 7, FINSEQ_2: 58;

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

      hence thesis;

    end;

    theorem :: MEASURE9:23

    

     Th21: for M be Matrix of ExtREAL st ( len M) = 0 holds ( SumAll M) = 0

    proof

      let M be Matrix of ExtREAL ;

      assume ( len M) = 0 ;

      then ( len ( Sum M)) = 0 by Def5;

      then ( Sum M) is empty;

      hence thesis by EXTREAL1: 7;

    end;

    theorem :: MEASURE9:24

    

     Th22: for m be Nat, M be Matrix of m, 0 , ExtREAL holds ( SumAll M) = 0

    proof

      let m be Nat, M be Matrix of m, 0 , ExtREAL ;

      per cases ;

        suppose m = 0 ;

        then ( len M) = 0 by MATRIX_0:def 2;

        hence thesis by Th21;

      end;

        suppose m > 0 ;

        then ( len M) > 0 by MATRIX_0:def 2;

        then

        reconsider k = ( len ( Sum M)) as non zero Nat by Def5;

        reconsider Z = 0. as R_eal;

        for k be Nat st k in ( dom ( Sum M)) holds (( Sum M) . k) = 0

        proof

          ( len M) = ( len ( Sum M)) by Def5;

          then

           A2: ( dom M) = ( dom ( Sum M)) by FINSEQ_3: 29;

          hereby

            let k be Nat;

            assume

             A3: k in ( dom ( Sum M));

            then (M . k) in ( rng M) by A2, FUNCT_1:def 3;

            then (M . k) = ( <*> ExtREAL ) by MATRIX_0:def 2;

            hence (( Sum M) . k) = 0 by A3, Def5, EXTREAL1: 7;

          end;

        end;

        then ( Sum M) = (k |-> 0. ) by MATRPROB: 1;

        then ( SumAll M) = (( len ( Sum M)) * Z) by Th20;

        hence ( SumAll M) = 0 ;

      end;

    end;

    theorem :: MEASURE9:25

    

     Th23: for n,m,k be Nat, M1 be Matrix of n, k, ExtREAL , M2 be Matrix of m, k, ExtREAL holds ( Sum (M1 ^ M2)) = (( Sum M1) ^ ( Sum M2))

    proof

      let n,m,k be Nat;

      let M1 be Matrix of n, k, ExtREAL ;

      let M2 be Matrix of m, k, ExtREAL ;

      

       A1: ( dom ( Sum (M1 ^ M2))) = ( Seg ( len ( Sum (M1 ^ M2)))) by FINSEQ_1:def 3;

       A2:

      now

        let i be Nat;

        assume

         A3: i in ( dom ( Sum (M1 ^ M2)));

        then i in ( Seg ( len (M1 ^ M2))) by A1, Def5;

        then

         A4: i in ( dom (M1 ^ M2)) by FINSEQ_1:def 3;

        

         A8: ( len M1) = ( len ( Sum M1)) & ( len M2) = ( len ( Sum M2)) by Def5;

        then

         A6: ( dom M1) = ( dom ( Sum M1)) & ( dom M2) = ( dom ( Sum M2)) by FINSEQ_3: 29;

        per cases by A4, FINSEQ_1: 25;

          suppose

           A5: i in ( dom M1);

          

          thus (( Sum (M1 ^ M2)) . i) = ( Sum ((M1 ^ M2) . i)) by A3, Def5

          .= ( Sum (M1 . i)) by A5, FINSEQ_1:def 7

          .= (( Sum M1) . i) by A5, A6, Def5

          .= ((( Sum M1) ^ ( Sum M2)) . i) by A5, A6, FINSEQ_1:def 7;

        end;

          suppose ex n be Nat st n in ( dom M2) & i = (( len M1) + n);

          then

          consider n be Nat such that

           A10: n in ( dom M2) & i = (( len M1) + n);

          

          thus (( Sum (M1 ^ M2)) . i) = ( Sum ((M1 ^ M2) . i)) by A3, Def5

          .= ( Sum (M2 . n)) by A10, FINSEQ_1:def 7

          .= (( Sum M2) . n) by A10, A6, Def5

          .= ((( Sum M1) ^ ( Sum M2)) . i) by A10, A8, A6, FINSEQ_1:def 7;

        end;

      end;

      ( len ( Sum (M1 ^ M2))) = ( len (M1 ^ M2)) by Def5

      .= (( len M1) + ( len M2)) by FINSEQ_1: 22

      .= (( len ( Sum M1)) + ( len M2)) by Def5

      .= (( len ( Sum M1)) + ( len ( Sum M2))) by Def5

      .= ( len (( Sum M1) ^ ( Sum M2))) by FINSEQ_1: 22;

      hence thesis by A2, FINSEQ_2: 9;

    end;

    theorem :: MEASURE9:26

    

     Th24: for M1,M2 be Matrix of ExtREAL st (for i be Nat st i in ( dom M1) holds not -infty in ( rng (M1 . i))) & (for i be Nat st i in ( dom M2) holds not -infty in ( rng (M2 . i))) holds (( Sum M1) + ( Sum M2)) = ( Sum (M1 ^^ M2))

    proof

      let M1,M2 be Matrix of ExtREAL ;

      reconsider M = ( min (( len M1),( len M2))) as Element of NAT ;

      assume

       B0: (for i be Nat st i in ( dom M1) holds not -infty in ( rng (M1 . i))) & (for i be Nat st i in ( dom M2) holds not -infty in ( rng (M2 . i)));

      now

        assume -infty in ( rng ( Sum M1));

        then

        consider i be Nat such that

         C1: i in ( dom ( Sum M1)) & (( Sum M1) . i) = -infty by FINSEQ_2: 10;

        i in ( Seg ( len ( Sum M1))) by C1, FINSEQ_1:def 3;

        then i in ( Seg ( len M1)) by Def5;

        then i in ( dom M1) by FINSEQ_1:def 3;

        then

         C2: not -infty in ( rng (M1 . i)) by B0;

        (( Sum M1) . i) = ( Sum (M1 . i)) by C1, Def5;

        then ex j be Nat st j in ( dom (M1 . i)) & ((M1 . i) . j) = -infty by C1, Th17;

        hence contradiction by C2, FUNCT_1: 3;

      end;

      then

       D1: (( Sum M1) " { -infty }) = {} by FUNCT_1: 72;

      now

        assume -infty in ( rng ( Sum M2));

        then

        consider i be Nat such that

         C1: i in ( dom ( Sum M2)) & (( Sum M2) . i) = -infty by FINSEQ_2: 10;

        i in ( Seg ( len ( Sum M2))) by C1, FINSEQ_1:def 3;

        then i in ( Seg ( len M2)) by Def5;

        then i in ( dom M2) by FINSEQ_1:def 3;

        then

         C2: not -infty in ( rng (M2 . i)) by B0;

        (( Sum M2) . i) = ( Sum (M2 . i)) by C1, Def5;

        then ex j be Nat st j in ( dom (M2 . i)) & ((M2 . i) . j) = -infty by C1, Th17;

        hence contradiction by C2, FUNCT_1: 3;

      end;

      then (( Sum M2) " { -infty }) = {} by FUNCT_1: 72;

      then

       D2: (( dom ( Sum M1)) /\ ( dom ( Sum M2))) = ((( dom ( Sum M1)) /\ ( dom ( Sum M2))) \ (((( Sum M1) " { -infty }) /\ (( Sum M2) " { +infty })) \/ ((( Sum M2) " { -infty }) /\ (( Sum M1) " { +infty })))) by D1;

      

       A1: ( Seg M) = (( Seg ( len M1)) /\ ( Seg ( len M2))) by FINSEQ_2: 2

      .= (( Seg ( len M1)) /\ ( dom M2)) by FINSEQ_1:def 3

      .= (( dom M1) /\ ( dom M2)) by FINSEQ_1:def 3

      .= ( dom (M1 ^^ M2)) by PRE_POLY:def 4

      .= ( Seg ( len (M1 ^^ M2))) by FINSEQ_1:def 3;

      

       A0: ( dom ( Sum M1)) = ( Seg ( len ( Sum M1))) & ( dom ( Sum M2)) = ( Seg ( len ( Sum M2))) by FINSEQ_1:def 3;

      ( dom (( Sum M1) + ( Sum M2))) = (( dom ( Sum M1)) /\ ( dom ( Sum M2))) by D2, MESFUNC1:def 3;

      then

       K1: ( dom (( Sum M1) + ( Sum M2))) = ( Seg ( min (( len ( Sum M1)),( len ( Sum M2))))) by A0, FINSEQ_2: 2;

      then

      reconsider SM12 = (( Sum M1) + ( Sum M2)) as FinSequence by FINSEQ_1:def 2;

      ( len SM12) = ( min (( len ( Sum M1)),( len ( Sum M2)))) by K1, FINSEQ_1:def 3;

      

      then

       A2: ( len SM12) = ( min (( len M1),( len ( Sum M2)))) by Def5

      .= ( min (( len M1),( len M2))) by Def5

      .= ( len (M1 ^^ M2)) by A1, FINSEQ_1: 6

      .= ( len ( Sum (M1 ^^ M2))) by Def5;

      

       A3: ( dom (( Sum M1) + ( Sum M2))) = ( Seg ( len SM12)) by FINSEQ_1:def 3;

      now

        let i be Nat;

        assume

         A4: i in ( dom (( Sum M1) + ( Sum M2)));

        then i in ( Seg ( len SM12)) by FINSEQ_1:def 3;

        then i in ( Seg ( len (M1 ^^ M2))) by A2, Def5;

        then

         A6: i in ( dom (M1 ^^ M2)) by FINSEQ_1:def 3;

        then i in (( dom M1) /\ ( dom M2)) by PRE_POLY:def 4;

        then

         B1: i in ( dom M1) & i in ( dom M2) by XBOOLE_0:def 4;

        then i in ( Seg ( len M1)) & i in ( Seg ( len M2)) by FINSEQ_1:def 3;

        then i in ( Seg ( len ( Sum M1))) & i in ( Seg ( len ( Sum M2))) by Def5;

        then

         A8: i in ( dom ( Sum M1)) & i in ( dom ( Sum M2)) by FINSEQ_1:def 3;

        

         A10: i in ( dom ( Sum (M1 ^^ M2))) by A2, A3, A4, FINSEQ_1:def 3;

        

         A11: ((M1 . i) ^ (M2 . i)) = ((M1 ^^ M2) . i) by A6, PRE_POLY:def 4;

        

         B3: not -infty in ( rng (M1 . i)) & not -infty in ( rng (M2 . i)) by B0, B1;

        

        thus ((( Sum M1) + ( Sum M2)) . i) = ((( Sum M1) . i) + (( Sum M2) . i)) by A4, MESFUNC1:def 3

        .= (( Sum (M1 . i)) + (( Sum M2) . i)) by A8, Def5

        .= (( Sum (M1 . i)) + ( Sum (M2 . i))) by A8, Def5

        .= ( Sum ((M1 ^^ M2) . i)) by A11, B3, EXTREAL1: 10

        .= (( Sum (M1 ^^ M2)) . i) by A10, Def5;

      end;

      hence thesis by A2, FINSEQ_2: 9;

    end;

    theorem :: MEASURE9:27

    

     Th25: for M1,M2 be Matrix of ExtREAL st ( len M1) = ( len M2) & (for i be Nat st i in ( dom M1) holds not -infty in ( rng (M1 . i))) & (for i be Nat st i in ( dom M2) holds not -infty in ( rng (M2 . i))) holds (( SumAll M1) + ( SumAll M2)) = ( SumAll (M1 ^^ M2))

    proof

      let M1,M2 be Matrix of ExtREAL such that

       A1: ( len M1) = ( len M2) & (for i be Nat st i in ( dom M1) holds not -infty in ( rng (M1 . i))) & (for i be Nat st i in ( dom M2) holds not -infty in ( rng (M2 . i)));

      

       A2: ( len ( Sum M1)) = ( len M1) by Def5

      .= ( len ( Sum M2)) by A1, Def5;

      then

      reconsider p1 = ( Sum M1), p2 = ( Sum M2) as Element of (( len ( Sum M1)) -tuples_on ExtREAL ) by FINSEQ_2: 92;

       C0:

      now

        assume -infty in ( rng ( Sum M1));

        then

        consider i be Nat such that

         C1: i in ( dom ( Sum M1)) & (( Sum M1) . i) = -infty by FINSEQ_2: 10;

        i in ( Seg ( len ( Sum M1))) by C1, FINSEQ_1:def 3;

        then i in ( Seg ( len M1)) by Def5;

        then i in ( dom M1) by FINSEQ_1:def 3;

        then

         C2: not -infty in ( rng (M1 . i)) by A1;

        (( Sum M1) . i) = ( Sum (M1 . i)) by C1, Def5;

        then ex j be Nat st j in ( dom (M1 . i)) & ((M1 . i) . j) = -infty by C1, Th17;

        hence contradiction by C2, FUNCT_1: 3;

      end;

       A3:

      now

        assume -infty in ( rng ( Sum M2));

        then

        consider i be Nat such that

         C1: i in ( dom ( Sum M2)) & (( Sum M2) . i) = -infty by FINSEQ_2: 10;

        i in ( Seg ( len ( Sum M2))) by C1, FINSEQ_1:def 3;

        then i in ( Seg ( len M2)) by Def5;

        then i in ( dom M2) by FINSEQ_1:def 3;

        then

         C2: not -infty in ( rng (M2 . i)) by A1;

        (( Sum M2) . i) = ( Sum (M2 . i)) by C1, Def5;

        then ex j be Nat st j in ( dom (M2 . i)) & ((M2 . i) . j) = -infty by C1, Th17;

        hence contradiction by C2, FUNCT_1: 3;

      end;

      

       A4: ( dom ( Sum M1)) = ( dom ( Sum M2)) by A2, FINSEQ_3: 29;

      ( Sum (M1 ^^ M2)) = (( Sum M1) + ( Sum M2)) by A1, Th24;

      hence (( SumAll M1) + ( SumAll M2)) = ( SumAll (M1 ^^ M2)) by A3, C0, A4, Th18;

    end;

    theorem :: MEASURE9:28

    

     Th26: for p be FinSequence of ExtREAL st not -infty in ( rng p) holds ( SumAll <*p*>) = ( SumAll ( <*p*> @ ))

    proof

      defpred x[ FinSequence of ExtREAL ] means not -infty in ( rng $1) implies ( SumAll <*$1*>) = ( SumAll ( <*$1*> @ ));

      let p be FinSequence of ExtREAL ;

      assume

       B0: not -infty in ( rng p);

      

       A2: for p be FinSequence of ExtREAL , x be Element of ExtREAL st x[p] holds x[(p ^ <*x*>)]

      proof

        let p be FinSequence of ExtREAL , x be Element of ExtREAL such that

         A3: not -infty in ( rng p) implies ( SumAll <*p*>) = ( SumAll ( <*p*> @ ));

        assume not -infty in ( rng (p ^ <*x*>));

        then not -infty in (( rng p) \/ ( rng <*x*>)) by FINSEQ_1: 31;

        then

         B3: not -infty in ( rng p) & not -infty in ( rng <*x*>) by XBOOLE_0:def 3;

        ( Seg ( len ( <*p*> ^^ <* <*x*>*>))) = ( dom ( <*p*> ^^ <* <*x*>*>)) by FINSEQ_1:def 3

        .= (( dom <*p*>) /\ ( dom <* <*x*>*>)) by PRE_POLY:def 4

        .= (( Seg 1) /\ ( dom <* <*x*>*>)) by FINSEQ_1: 38

        .= (( Seg 1) /\ ( Seg 1)) by FINSEQ_1: 38

        .= ( Seg 1);

        

        then

         A4: ( len ( <*p*> ^^ <* <*x*>*>)) = 1 by FINSEQ_1: 6

        .= ( len <*(p ^ <*x*>)*>) by FINSEQ_1: 39;

        

         A5: ( dom <*(p ^ <*x*>)*>) = ( Seg ( len <*(p ^ <*x*>)*>)) by FINSEQ_1:def 3;

         A6:

        now

          let i be Nat;

          reconsider M1 = ( <*p*> . i), M2 = ( <* <*x*>*> . i) as FinSequence;

          assume

           A7: i in ( dom <*(p ^ <*x*>)*>);

          then

           A8: i = 1 by FINSEQ_1: 90;

          i in ( dom ( <*p*> ^^ <* <*x*>*>)) by A4, A5, A7, FINSEQ_1:def 3;

          

          hence (( <*p*> ^^ <* <*x*>*>) . i) = (M1 ^ M2) by PRE_POLY:def 4

          .= (p ^ M2) by A8, FINSEQ_1: 40

          .= (p ^ <*x*>) by A8, FINSEQ_1: 40

          .= ( <*(p ^ <*x*>)*> . i) by A8, FINSEQ_1: 40;

        end;

        per cases ;

          suppose ( len p) = 0 ;

          then

           A9: p = {} ;

          

          hence ( SumAll <*(p ^ <*x*>)*>) = ( SumAll <* <*x*>*>) by FINSEQ_1: 34

          .= ( SumAll ( <* <*x*>*> @ )) by MATRLIN: 15

          .= ( SumAll ( <*(p ^ <*x*>)*> @ )) by A9, FINSEQ_1: 34;

        end;

          suppose

           A10: ( len p) <> 0 ;

          

           A11: ( len <* <*x*>*>) = 1 & ( len <*p*>) = 1 & ( len <*x*>) = 1 by FINSEQ_1: 40;

          then

           A12: ( width <* <*x*>*>) = 1 & ( width <*p*>) = ( len p) by MATRIX_0: 20;

          then

           A16: ( len ( <*p*> @ )) = ( len p) by MATRIX_0:def 6;

          

           P5: ( width ( <*p*> @ )) = 1 by A10, A11, A12, MATRIX_0: 29;

          then

          reconsider d1 = ( <*p*> @ ) as Matrix of ( len p), 1, ExtREAL by A10, A16, MATRIX_0: 20;

          

           A13: ( len ( <* <*x*>*> @ )) = 1 by A12, MATRIX_0: 54;

          

           PP5: ( width ( <* <*x*>*> @ )) = 1 by A11, A12, MATRIX_0: 29;

          then

          reconsider d2 = ( <* <*x*>*> @ ) as Matrix of 1, 1, ExtREAL by A13, MATRIX_0: 20;

          ( len <*(p ^ <*x*>)*>) = 1 by FINSEQ_1: 40;

          

          then

           A18: ( width <*(p ^ <*x*>)*>) = ( len (p ^ <*x*>)) by MATRIX_0: 20

          .= (( len p) + 1) by A11, FINSEQ_1: 22;

          

           A19: (( <* <*x*>*> @ ) @ ) = <* <*x*>*> by A11, A12, MATRIX_0: 57;

          ( width ( <*p*> @ )) = ( width ( <* <*x*>*> @ )) by P5, A11, A12, MATRIX_0: 29;

          

          then

           A21: ((d1 ^ d2) @ ) = ((( <*p*> @ ) @ ) ^^ (( <* <*x*>*> @ ) @ )) by MATRLIN: 28

          .= ( <*p*> ^^ <* <*x*>*>) by A10, A11, A12, A19, MATRIX_0: 57

          .= <*(p ^ <*x*>)*> by A4, A6, FINSEQ_2: 9

          .= (( <*(p ^ <*x*>)*> @ ) @ ) by A18, MATRIX_0: 57;

          

           A22: ( len (( <*p*> @ ) ^ ( <* <*x*>*> @ ))) = (( len ( <*p*> @ )) + ( len ( <* <*x*>*> @ ))) by FINSEQ_1: 22

          .= (( width <*p*>) + ( len ( <* <*x*>*> @ ))) by MATRIX_0:def 6

          .= (( width <*p*>) + ( width <* <*x*>*>)) by MATRIX_0:def 6

          .= ( len ( <*(p ^ <*x*>)*> @ )) by A12, A18, MATRIX_0:def 6;

           B4:

          now

            let i be Nat;

            assume i in ( dom <*p*>);

            then i = 1 by FINSEQ_1: 90;

            hence not -infty in ( rng ( <*p*> . i)) by B3, FINSEQ_1:def 8;

          end;

           B5:

          now

            let i be Nat;

            assume i in ( dom <* <*x*>*>);

            then i = 1 by FINSEQ_1: 90;

            hence not -infty in ( rng ( <* <*x*>*> . i)) by B3, FINSEQ_1:def 8;

          end;

          ( dom <*p*>) = ( Seg 1) by FINSEQ_1: 38;

          then 1 in ( dom <*p*>);

          then

           B6: not -infty in ( rng ( <*p*> . 1)) by B4;

          then

           T6: not -infty in ( rng p) by FINSEQ_1:def 8;

          for x be object st x in ( dom ( Sum d1)) holds (( Sum d1) . x) <> -infty

          proof

            let x be object;

            assume

             P1: x in ( dom ( Sum d1));

            then

            reconsider i = x as Nat;

            

             P2: (( Sum d1) . x) = ( Sum (d1 . i)) by P1, Def5;

            1 <= i & i <= ( len ( Sum d1)) by P1, FINSEQ_3: 25;

            then

             P3: 1 <= i & i <= ( len d1) by Def5;

            then i in ( dom p) by A16, FINSEQ_3: 25;

            then

             R10: (p . i) <> -infty by T6, FUNCT_1: 3;

            i in ( dom d1) by P3, FINSEQ_3: 25;

            then

             P4: (d1 . i) = ( Line (d1,i)) by MATRIX_0: 60;

            ( dom d1) = ( Seg ( len p)) by A16, FINSEQ_1:def 3;

            then

             R2: ( Indices d1) = [:( Seg ( len p)), {1}:] by P5, FINSEQ_1: 2, MATRIX_0:def 4;

            

             R3: i in ( Seg ( len p)) by P3, A16;

            for j be Nat st j in ( dom ( Line (d1,i))) holds (( Line (d1,i)) . j) <> -infty

            proof

              let j be Nat;

              assume j in ( dom ( Line (d1,i)));

              then 1 <= j & j <= ( len ( Line (d1,i))) by FINSEQ_3: 25;

              then 1 <= j & j <= ( width d1) by MATRIX_0:def 7;

              then

               P6: j in ( Seg ( width d1));

              then

               R4: [i, j] in [:( Seg ( len p)), {1}:] by P5, R3, FINSEQ_1: 2, ZFMISC_1:def 2;

              then [j, i] in ( Indices (d1 @ )) by R2, MATRIX_0:def 6;

              then

              consider F be FinSequence of ExtREAL such that

               R7: F = ((d1 @ ) . j) & ((d1 @ ) * (j,i)) = (F . i) by MATRIX_0:def 5;

              F = ( <*p*> . j) by A10, A12, R7, A11, MATRIX_0: 57;

              then F = ( <*p*> . 1) by P5, P6, FINSEQ_1: 2, TARSKI:def 1;

              then

               R9: F = p by FINSEQ_1:def 8;

              (( Line (d1,i)) . j) = (( <*p*> @ ) * (i,j)) by P6, MATRIX_0:def 7;

              hence (( Line (d1,i)) . j) <> -infty by R7, R9, R10, R2, R4, MATRIX_0:def 6;

            end;

            hence (( Sum d1) . x) <> -infty by P2, P4, Th17;

          end;

          then

           B7: not -infty in ( rng ( Sum d1)) by FUNCT_1:def 3;

          for z be object st z in ( dom ( Sum d2)) holds (( Sum d2) . z) <> -infty

          proof

            let z be object;

            assume

             P1: z in ( dom ( Sum d2));

            then

            reconsider i = z as Nat;

            

             P2: (( Sum d2) . z) = ( Sum (d2 . i)) by P1, Def5;

            1 <= i & i <= ( len ( Sum d2)) by P1, FINSEQ_3: 25;

            then

             P3: 1 <= i & i <= ( len d2) by Def5;

            then

             R1: 1 <= i & i <= ( len <*x*>) by A13, FINSEQ_1: 40;

            then i in ( dom <*x*>) by FINSEQ_3: 25;

            then

             R10: ( <*x*> . i) <> -infty by B3, FUNCT_1: 3;

            i in ( dom d2) by P3, FINSEQ_3: 25;

            then

             P4: (d2 . i) = ( Line (d2,i)) by MATRIX_0: 60;

            ( dom d2) = ( Seg ( len <*x*>)) by A13, FINSEQ_1:def 3, FINSEQ_1: 40;

            then

             R2: ( Indices d2) = [:( Seg ( len <*x*>)), {1}:] by PP5, FINSEQ_1: 2, MATRIX_0:def 4;

            

             R3: i in ( Seg ( len <*x*>)) by R1;

            for j be Nat st j in ( dom ( Line (d2,i))) holds (( Line (d2,i)) . j) <> -infty

            proof

              let j be Nat;

              assume j in ( dom ( Line (d2,i)));

              then 1 <= j & j <= ( len ( Line (d2,i))) by FINSEQ_3: 25;

              then 1 <= j & j <= ( width d2) by MATRIX_0:def 7;

              then

               P6: j in ( Seg ( width d2));

              then

               R4: [i, j] in [:( Seg ( len <*x*>)), {1}:] by PP5, R3, FINSEQ_1: 2, ZFMISC_1:def 2;

              then [j, i] in ( Indices (d2 @ )) by R2, MATRIX_0:def 6;

              then

              consider F be FinSequence of ExtREAL such that

               R7: F = ((d2 @ ) . j) & ((d2 @ ) * (j,i)) = (F . i) by MATRIX_0:def 5;

              F = ( <* <*x*>*> . j) by A12, R7, A11, MATRIX_0: 57;

              then F = ( <* <*x*>*> . 1) by PP5, P6, FINSEQ_1: 2, TARSKI:def 1;

              then

               R9: F = <*x*> by FINSEQ_1:def 8;

              (( Line (d2,i)) . j) = (( <* <*x*>*> @ ) * (i,j)) by P6, MATRIX_0:def 7;

              hence (( Line (d2,i)) . j) <> -infty by R7, R9, R10, R2, R4, MATRIX_0:def 6;

            end;

            hence (( Sum d2) . z) <> -infty by P2, P4, Th17;

          end;

          then

           B8: not -infty in ( rng ( Sum d2)) by FUNCT_1:def 3;

          

          thus ( SumAll <*(p ^ <*x*>)*>) = ( SumAll ( <*p*> ^^ <* <*x*>*>)) by A4, A6, FINSEQ_2: 9

          .= (( SumAll <*p*>) + ( SumAll <* <*x*>*>)) by A11, B4, B5, Th25

          .= (( SumAll ( <*p*> @ )) + ( SumAll ( <* <*x*>*> @ ))) by A3, B6, FINSEQ_1:def 8, MATRLIN: 15

          .= ( Sum (( Sum d1) ^ ( Sum d2))) by B7, B8, EXTREAL1: 10

          .= ( SumAll (d1 ^ d2)) by Th23

          .= ( SumAll ( <*(p ^ <*x*>)*> @ )) by A22, A21, MATRIX_0: 53;

        end;

      end;

      

       A23: x[( <*> ExtREAL )]

      proof

        reconsider M1 = <*( <*> ExtREAL )*> as Matrix of 1, 0 , ExtREAL by MATRIX_0: 14;

        ( len M1) = 1 by MATRIX_0:def 2;

        then ( width M1) = 0 by MATRIX_0: 20;

        then

         A24: ( len (M1 @ )) = 0 by MATRIX_0:def 6;

        ( SumAll M1) = 0 by Th22;

        hence thesis by A24, Th21;

      end;

      for p be FinSequence of ExtREAL holds x[p] from FINSEQ_2:sch 2( A23, A2);

      hence thesis by B0;

    end;

    theorem :: MEASURE9:29

    

     Th27: for p be ext-real number, M be Matrix of ExtREAL st (for i be Nat st i in ( dom M) holds not p in ( rng (M . i))) holds (for j be Nat st j in ( dom (M @ )) holds not p in ( rng ((M @ ) . j)))

    proof

      let p be ext-real number;

      let M be Matrix of ExtREAL ;

      assume

       A1: for i be Nat st i in ( dom M) holds not p in ( rng (M . i));

      hereby

        let j be Nat;

        assume

         A2: j in ( dom (M @ ));

        then

         A3: ((M @ ) . j) = ( Line ((M @ ),j)) by MATRIX_0: 60;

        j in ( Seg ( len (M @ ))) by A2, FINSEQ_1:def 3;

        then j in ( Seg ( width M)) by MATRIX_0:def 6;

        then

         A5: ( Line ((M @ ),j)) = ( Col (M,j)) by MATRIX_0: 59;

        for v be object st v in ( dom ( Line ((M @ ),j))) holds (( Line ((M @ ),j)) . v) <> p

        proof

          let v be object;

          assume

           A6: v in ( dom ( Line ((M @ ),j)));

          then

          reconsider i = v as Element of NAT ;

          1 <= i & i <= ( len ( Line ((M @ ),j))) by A6, FINSEQ_3: 25;

          then 1 <= i & i <= ( width (M @ )) by MATRIX_0:def 7;

          then i in ( Seg ( width (M @ )));

          then [j, i] in [:( dom (M @ )), ( Seg ( width (M @ ))):] by A2, ZFMISC_1:def 2;

          then [j, i] in ( Indices (M @ )) by MATRIX_0:def 4;

          then

           A7: [i, j] in ( Indices M) by MATRIX_0:def 6;

          then

           A8: i in ( dom M) & j in ( dom (M . i)) by MATRPROB: 13;

          then (( Line ((M @ ),j)) . v) = (M * (i,j)) by A5, MATRIX_0:def 8;

          then (( Line ((M @ ),j)) . v) = ((M . i) . j) by A7, MATRPROB: 14;

          then (( Line ((M @ ),j)) . v) in ( rng (M . i)) by A8, FUNCT_1: 3;

          hence (( Line ((M @ ),j)) . v) <> p by A1, A7, MATRPROB: 13;

        end;

        hence not p in ( rng ((M @ ) . j)) by A3, FUNCT_1:def 3;

      end;

    end;

    theorem :: MEASURE9:30

    

     Th28: for M be Matrix of ExtREAL st (for i be Nat st i in ( dom M) holds not -infty in ( rng (M . i))) holds ( SumAll M) = ( SumAll (M @ ))

    proof

      let M be Matrix of ExtREAL ;

      assume

       A0: for i be Nat st i in ( dom M) holds not -infty in ( rng (M . i));

      defpred x[ Nat] means for M be Matrix of ExtREAL st ( len M) = $1 & (for i be Nat st i in ( dom M) holds not -infty in ( rng (M . i))) holds ( SumAll M) = ( SumAll (M @ ));

      

       A1: for n be Nat st x[n] holds x[(n + 1)]

      proof

        let n be Nat;

        assume

         A2: for M be Matrix of ExtREAL st ( len M) = n & (for i be Nat st i in ( dom M) holds not -infty in ( rng (M . i))) holds ( SumAll M) = ( SumAll (M @ ));

        thus for M be Matrix of ExtREAL st ( len M) = (n + 1) & (for i be Nat st i in ( dom M) holds not -infty in ( rng (M . i))) holds ( SumAll M) = ( SumAll (M @ ))

        proof

          let M be Matrix of ExtREAL ;

          assume

           A3: ( len M) = (n + 1) & (for i be Nat st i in ( dom M) holds not -infty in ( rng (M . i)));

          then

           a3: M <> {} ;

          per cases ;

            suppose

             A4: n = 0 ;

            1 <= ( len M) by A3, NAT_1: 11;

            then

             A5: not -infty in ( rng (M . 1)) by A3, FINSEQ_3: 25;

            M = <*(M . 1)*> by A3, A4, FINSEQ_1: 40;

            hence thesis by A5, Th26;

          end;

            suppose

             A30: n > 0 ;

            reconsider M9 = M as Matrix of (n + 1), ( width M), ExtREAL by A3, MATRIX_0: 20;

            reconsider M1 = (M . (n + 1)) as FinSequence of ExtREAL ;

            reconsider w = ( Del (M9,(n + 1))) as Matrix of n, ( width M), ExtREAL by MATRLIN: 3;

            

             V1: 1 <= (n + 1) by NAT_1: 11;

            then (M . (n + 1)) = ( Line (M,(n + 1))) by A3, FINSEQ_3: 25, MATRIX_0: 60;

            then

             Y11: ( len M1) = ( width M) by MATRIX_0:def 7;

            then

            reconsider r = <*M1*> as Matrix of 1, ( width M), ExtREAL ;

            

             A31: ( width w) = ( width M9) by A30, MATRLIN: 2

            .= ( width r) by MATRLIN: 2;

            

             A32: ( len (w @ )) = ( width w) by MATRIX_0:def 6

            .= ( len (r @ )) by A31, MATRIX_0:def 6;

            

             A33: ( len ( Del (M,(n + 1)))) = n by A3, PRE_POLY: 12;

            

             T5: not -infty in ( rng M1) by V1, A3, FINSEQ_3: 25;

            for v be object st v in ( dom ( Sum w)) holds (( Sum w) . v) <> -infty

            proof

              let v be object;

              assume

               P1: v in ( dom ( Sum w));

              then

              reconsider i = v as Nat;

              

               P2: (( Sum w) . v) = ( Sum (w . i)) by P1, Def5;

              1 <= i & i <= ( len ( Sum w)) by P1, FINSEQ_3: 25;

              then

               P3: 1 <= i & i <= ( len w) by Def5;

              then

               S0: 1 <= i & i <= (n + 1) by A33, NAT_1: 12;

              

               R1: i in ( dom w) by P3, FINSEQ_3: 25;

              then

               P4: (w . i) = ( Line (w,i)) by MATRIX_0: 60;

              for j be Nat st j in ( dom ( Line (w,i))) holds (( Line (w,i)) . j) <> -infty

              proof

                let j be Nat;

                assume j in ( dom ( Line (w,i)));

                then 1 <= j & j <= ( len ( Line (w,i))) by FINSEQ_3: 25;

                then 1 <= j & j <= ( width w) by MATRIX_0:def 7;

                then

                 P6: j in ( Seg ( width w));

                then [i, j] in [:( dom w), ( Seg ( width w)):] by R1, ZFMISC_1:def 2;

                then [i, j] in ( Indices w) by MATRIX_0:def 4;

                then

                consider F be FinSequence of ExtREAL such that

                 R7: F = (w . i) & (w * (i,j)) = (F . j) by MATRIX_0:def 5;

                M <> {} by A3;

                then M = (( Del (M,( len M))) ^ <*(M . ( len M))*>) by PRE_POLY: 13;

                then (M . i) = (w . i) by A3, R1, FINSEQ_1:def 7;

                then

                 S2: not -infty in ( rng F) by R7, A3, S0, FINSEQ_3: 25;

                ( len F) = ( width w) by P4, R7, MATRIX_0:def 7;

                then j in ( dom F) by P6, FINSEQ_1:def 3;

                then (F . j) in ( rng F) by FUNCT_1: 3;

                hence (( Line (w,i)) . j) <> -infty by R7, S2, P6, MATRIX_0:def 7;

              end;

              hence (( Sum w) . v) <> -infty by P2, P4, Th17;

            end;

            then

             L1: not -infty in ( rng ( Sum w)) by FUNCT_1:def 3;

            for v be object st v in ( dom ( Sum r)) holds (( Sum r) . v) <> -infty

            proof

              let v be object;

              assume

               P1: v in ( dom ( Sum r));

              then

              reconsider i = v as Nat;

              

               P2: (( Sum r) . v) = ( Sum (r . i)) by P1, Def5;

              1 <= i & i <= ( len ( Sum r)) by P1, FINSEQ_3: 25;

              then

               P3: 1 <= i & i <= ( len r) by Def5;

              then 1 <= i & i <= 1 by FINSEQ_1: 40;

              then i = 1 by XXREAL_0: 1;

              then (n + i) in ( Seg (n + 1)) by FINSEQ_1: 4;

              then

               S0: (n + i) in ( dom M) by A3, FINSEQ_1:def 3;

              

               R1: i in ( dom r) by P3, FINSEQ_3: 25;

              then

               P4: (r . i) = ( Line (r,i)) by MATRIX_0: 60;

              for j be Nat st j in ( dom ( Line (r,i))) holds (( Line (r,i)) . j) <> -infty

              proof

                let j be Nat;

                assume j in ( dom ( Line (r,i)));

                then 1 <= j & j <= ( len ( Line (r,i))) by FINSEQ_3: 25;

                then 1 <= j & j <= ( width r) by MATRIX_0:def 7;

                then

                 P6: j in ( Seg ( width r));

                then [i, j] in [:( dom r), ( Seg ( width r)):] by R1, ZFMISC_1:def 2;

                then [i, j] in ( Indices r) by MATRIX_0:def 4;

                then

                consider F be FinSequence of ExtREAL such that

                 R7: F = (r . i) & (r * (i,j)) = (F . j) by MATRIX_0:def 5;

                M <> {} by A3;

                then M = (w ^ <*(M . (n + 1))*>) by A3, PRE_POLY: 13;

                then (M . (n + i)) = (r . i) by A33, R1, FINSEQ_1:def 7;

                then

                 S2: not -infty in ( rng F) by R7, A3, S0;

                ( len F) = ( width r) by P4, R7, MATRIX_0:def 7;

                then j in ( dom F) by P6, FINSEQ_1:def 3;

                then (F . j) in ( rng F) by FUNCT_1: 3;

                hence (( Line (r,i)) . j) <> -infty by R7, S2, P6, MATRIX_0:def 7;

              end;

              hence (( Sum r) . v) <> -infty by P2, P4, Th17;

            end;

            then

             T3: not -infty in ( rng ( Sum r)) by FUNCT_1:def 3;

            

             T4: for i be Nat st i in ( dom ( Del (M,(n + 1)))) holds not -infty in ( rng (( Del (M,(n + 1))) . i))

            proof

              let i be Nat;

              assume

               R1: i in ( dom ( Del (M,(n + 1))));

              then

               P4: (w . i) = ( Line (w,i)) by MATRIX_0: 60;

              1 <= i & i <= ( len w) by R1, FINSEQ_3: 25;

              then

               S0: 1 <= i & i <= (n + 1) by A33, NAT_1: 12;

              for v be object st v in ( dom ( Line (w,i))) holds (( Line (w,i)) . v) <> -infty

              proof

                let v be object;

                assume

                 TT0: v in ( dom ( Line (w,i)));

                then

                reconsider j = v as Nat;

                1 <= j & j <= ( len ( Line (w,i))) by TT0, FINSEQ_3: 25;

                then 1 <= j & j <= ( width w) by MATRIX_0:def 7;

                then

                 P6: j in ( Seg ( width w));

                then [i, j] in [:( dom w), ( Seg ( width w)):] by R1, ZFMISC_1:def 2;

                then [i, j] in ( Indices w) by MATRIX_0:def 4;

                then

                consider F be FinSequence of ExtREAL such that

                 R7: F = (w . i) & (w * (i,j)) = (F . j) by MATRIX_0:def 5;

                M <> {} by A3;

                then M = (( Del (M,( len M))) ^ <*(M . ( len M))*>) by PRE_POLY: 13;

                then (M . i) = (w . i) by A3, R1, FINSEQ_1:def 7;

                then

                 S2: not -infty in ( rng F) by R7, A3, S0, FINSEQ_3: 25;

                ( len F) = ( width w) by P4, R7, MATRIX_0:def 7;

                then j in ( dom F) by P6, FINSEQ_1:def 3;

                then (F . j) in ( rng F) by FUNCT_1: 3;

                hence (( Line (w,i)) . v) <> -infty by R7, S2, P6, MATRIX_0:def 7;

              end;

              hence not -infty in ( rng (( Del (M,(n + 1))) . i)) by P4, FUNCT_1:def 3;

            end;

            M <> {} by A3;

            then M = (( Del (M,( len M))) ^ <*(M . ( len M))*>) by PRE_POLY: 13;

            then

             H1: (M @ ) = ((w @ ) ^^ ( <*(M . (n + 1))*> @ )) by A3, A31, MATRLIN: 28;

            then

             Q4: ( dom (M @ )) = (( dom (w @ )) /\ ( dom ( <*(M . (n + 1))*> @ ))) by PRE_POLY:def 4;

            ( dom (w @ )) = ( Seg ( len (w @ ))) by FINSEQ_1:def 3;

            then ( dom (w @ )) = ( Seg ( width w)) by MATRIX_0:def 6;

            then ( dom (w @ )) = ( Seg ( len ( <*(M . (n + 1))*> @ ))) by A31, MATRIX_0:def 6;

            then

             Z0: ( dom (w @ )) = ( dom ( <*(M . (n + 1))*> @ )) by FINSEQ_1:def 3;

            

             Y2: ( len <*(M . (n + 1))*>) = 1 by FINSEQ_1: 40;

            then

             Z2: ( width <*(M . (n + 1))*>) = ( width M) by Y11, MATRIX_0: 20;

            

             T6: for i be Nat st i in ( dom (w @ )) holds not -infty in ( rng ((w @ ) . i))

            proof

              let i be Nat;

              assume

               R1: i in ( dom (w @ ));

              then

               P4: ((w @ ) . i) = ( Line ((w @ ),i)) by MATRIX_0: 60;

              1 <= i & i <= ( len (w @ )) by R1, FINSEQ_3: 25;

              then 1 <= i & i <= ( width w) by MATRIX_0:def 6;

              then 1 <= ( width w) by XXREAL_0: 2;

              then

               V5: 1 <= ( width M) by A30, MATRLIN: 2;

              for v be object st v in ( dom ( Line ((w @ ),i))) holds (( Line ((w @ ),i)) . v) <> -infty

              proof

                let v be object;

                assume

                 TT0: v in ( dom ( Line ((w @ ),i)));

                then

                reconsider j = v as Nat;

                1 <= j & j <= ( len ( Line ((w @ ),i))) by TT0, FINSEQ_3: 25;

                then 1 <= j & j <= ( width (w @ )) by MATRIX_0:def 7;

                then

                 P6: j in ( Seg ( width (w @ )));

                then [i, j] in [:( dom (w @ )), ( Seg ( width (w @ ))):] by R1, ZFMISC_1:def 2;

                then [i, j] in ( Indices (w @ )) by MATRIX_0:def 4;

                then

                consider F be FinSequence of ExtREAL such that

                 R7: F = ((w @ ) . i) & ((w @ ) * (i,j)) = (F . j) by MATRIX_0:def 5;

                ( width ( <*(M . (n + 1))*> @ )) = ( len <*(M . (n + 1))*>) by V5, Z2, MATRIX_0: 29;

                then 1 in ( Seg ( width ( <*(M . (n + 1))*> @ ))) by Y2;

                then [i, 1] in [:( dom ( <*(M . (n + 1))*> @ )), ( Seg ( width ( <*(M . (n + 1))*> @ ))):] by Z0, R1, ZFMISC_1: 87;

                then [i, 1] in ( Indices ( <*(M . (n + 1))*> @ )) by MATRIX_0:def 4;

                then

                consider G be FinSequence of ExtREAL such that

                 Q7: G = (( <*(M . (n + 1))*> @ ) . i) & (( <*(M . (n + 1))*> @ ) * (i,1)) = (G . 1) by MATRIX_0:def 5;

                ((M @ ) . i) = (F ^ G) by R7, H1, Z0, Q4, R1, Q7, PRE_POLY:def 4;

                then not -infty in ( rng (F ^ G)) by Z0, Q4, R1, A3, Th27;

                then not -infty in (( rng F) \/ ( rng G)) by FINSEQ_1: 31;

                then

                 S2: not -infty in ( rng F) & not -infty in ( rng G) by XBOOLE_0:def 3;

                ( len F) = ( width (w @ )) by P4, R7, MATRIX_0:def 7;

                then j in ( dom F) by P6, FINSEQ_1:def 3;

                then (F . j) in ( rng F) by FUNCT_1: 3;

                hence (( Line ((w @ ),i)) . v) <> -infty by R7, S2, P6, MATRIX_0:def 7;

              end;

              hence not -infty in ( rng ((w @ ) . i)) by P4, FUNCT_1:def 3;

            end;

            

             T7: for i be Nat st i in ( dom (r @ )) holds not -infty in ( rng ((r @ ) . i))

            proof

              let i be Nat;

              assume

               R1: i in ( dom (r @ ));

              then

               P4: ((r @ ) . i) = ( Line ((r @ ),i)) by MATRIX_0: 60;

              1 <= i & i <= ( len (r @ )) by R1, FINSEQ_3: 25;

              then 1 <= i & i <= ( width r) by MATRIX_0:def 6;

              then 1 <= ( width r) by XXREAL_0: 2;

              then

               M1: 1 <= ( width M9) by MATRLIN: 2;

              for v be object st v in ( dom ( Line ((r @ ),i))) holds (( Line ((r @ ),i)) . v) <> -infty

              proof

                let v be object;

                assume

                 TT0: v in ( dom ( Line ((r @ ),i)));

                then

                reconsider j = v as Nat;

                1 <= j & j <= ( len ( Line ((r @ ),i))) by TT0, FINSEQ_3: 25;

                then 1 <= j & j <= ( width (r @ )) by MATRIX_0:def 7;

                then

                 P6: j in ( Seg ( width (r @ )));

                then [i, j] in [:( dom (r @ )), ( Seg ( width (r @ ))):] by R1, ZFMISC_1:def 2;

                then [i, j] in ( Indices (r @ )) by MATRIX_0:def 4;

                then

                consider G be FinSequence of ExtREAL such that

                 R7: G = ((r @ ) . i) & ((r @ ) * (i,j)) = (G . j) by MATRIX_0:def 5;

                1 <= ( width w) by A30, M1, MATRLIN: 2;

                then ( width (w @ )) = ( len w) by MATRIX_0: 29;

                then n in ( Seg ( width (w @ ))) by A30, A33, FINSEQ_1: 3;

                then [i, n] in [:( dom (w @ )), ( Seg ( width (w @ ))):] by Z0, R1, ZFMISC_1: 87;

                then [i, n] in ( Indices (w @ )) by MATRIX_0:def 4;

                then

                consider F be FinSequence of ExtREAL such that

                 Q7: F = ((w @ ) . i) & ((w @ ) * (i,n)) = (F . n) by MATRIX_0:def 5;

                ((M @ ) . i) = (F ^ G) by R7, H1, Z0, Q4, R1, Q7, PRE_POLY:def 4;

                then not -infty in ( rng (F ^ G)) by Z0, Q4, R1, A3, Th27;

                then not -infty in (( rng F) \/ ( rng G)) by FINSEQ_1: 31;

                then

                 S2: not -infty in ( rng F) & not -infty in ( rng G) by XBOOLE_0:def 3;

                ( len G) = ( width (r @ )) by P4, R7, MATRIX_0:def 7;

                then j in ( dom G) by P6, FINSEQ_1:def 3;

                then (G . j) in ( rng G) by FUNCT_1: 3;

                hence (( Line ((r @ ),i)) . v) <> -infty by S2, R7, P6, MATRIX_0:def 7;

              end;

              hence not -infty in ( rng ((r @ ) . i)) by P4, FUNCT_1:def 3;

            end;

            

            thus ( SumAll M) = ( SumAll (w ^ r)) by A3, PRE_POLY: 13, a3

            .= ( Sum (( Sum w) ^ ( Sum r))) by Th23

            .= (( SumAll ( Del (M,(n + 1)))) + ( SumAll r)) by T3, L1, EXTREAL1: 10

            .= (( SumAll (( Del (M,(n + 1))) @ )) + ( SumAll r)) by A2, A33, T4

            .= (( SumAll (( Del (M,(n + 1))) @ )) + ( SumAll (r @ ))) by T5, Th26

            .= ( SumAll ((w @ ) ^^ (r @ ))) by A32, Th25, T6, T7

            .= ( SumAll ((w ^ r) @ )) by A31, MATRLIN: 28

            .= ( SumAll (M @ )) by A3, PRE_POLY: 13, a3;

          end;

        end;

      end;

      

       A34: x[ 0 ]

      proof

        let M be Matrix of ExtREAL ;

        assume

         A35: ( len M) = 0 & (for i be Nat st i in ( dom M) holds not -infty in ( rng (M . i)));

        then ( width M) = 0 by MATRIX_0:def 3;

        then

         A36: ( len (M @ )) = 0 by MATRIX_0:def 6;

        

        thus ( SumAll M) = 0 by A35, Th21

        .= ( SumAll (M @ )) by A36, Th21;

      end;

      for n be Nat holds x[n] from NAT_1:sch 2( A34, A1);

      then x[( len M)];

      hence thesis by A0;

    end;

    begin

    registration

      let x be object;

      cluster <*x*> -> disjoint_valued;

      correctness

      proof

        now

          let i,j be object;

          assume

           A3: i <> j;

          per cases ;

            suppose i in ( dom <*x*>);

            then i in {1} by FINSEQ_1: 2, FINSEQ_1: 38;

            then i = 1 by TARSKI:def 1;

            then not j in {1} by A3, TARSKI:def 1;

            then not j in ( dom <*x*>) by FINSEQ_1: 2, FINSEQ_1: 38;

            then ( <*x*> . j) = {} by FUNCT_1:def 2;

            hence ( <*x*> . i) misses ( <*x*> . j) by XBOOLE_1: 65;

          end;

            suppose not i in ( dom <*x*>);

            then ( <*x*> . i) = {} by FUNCT_1:def 2;

            hence ( <*x*> . i) misses ( <*x*> . j) by XBOOLE_1: 65;

          end;

        end;

        hence <*x*> is disjoint_valued by PROB_2:def 2;

      end;

    end

    theorem :: MEASURE9:31

    for X be set, S be with_empty_element semi-diff-closed cap-closed Subset-Family of X, F be FinSequence of S, G be Element of S holds ex H be disjoint_valued FinSequence of S st (G \ ( Union F)) = ( Union H)

    proof

      let X be set, S be with_empty_element semi-diff-closed cap-closed Subset-Family of X, F be FinSequence of S, G be Element of S;

      defpred P[ Nat] means for f be FinSequence of S st ( len f) = $1 holds ex H be disjoint_valued FinSequence of S st (G \ ( Union f)) = ( Union H);

      for f be FinSequence of S st ( len f) = 0 holds ex H be disjoint_valued FinSequence of S st (G \ ( Union f)) = ( Union H)

      proof

        let f be FinSequence of S;

        assume ( len f) = 0 ;

        then f = {} ;

        then ( rng f) = {} ;

        then

         A4: ( Union f) = {} by CARD_3:def 4, ZFMISC_1: 2;

        

         A5: ( rng <*G*>) = {G} by FINSEQ_1: 38;

        reconsider H = <*G*> as disjoint_valued FinSequence of S;

        take H;

        ( union ( rng H)) = G by A5, ZFMISC_1: 25;

        hence (G \ ( Union f)) = ( Union H) by A4, CARD_3:def 4;

      end;

      then

       A6: P[ 0 ];

      

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

      proof

        let i be Nat;

        assume

         A8: P[i];

        now

          let f be FinSequence of S;

          assume

           A9: ( len f) = (i + 1);

          then ( len (f | i)) = i by NAT_1: 11, FINSEQ_1: 59;

          then

          consider h be disjoint_valued FinSequence of S such that

           A12: (G \ ( Union (f | i))) = ( Union h) by A8;

          

           A10: f = ((f | i) ^ <*(f . (i + 1))*>) by A9, FINSEQ_3: 55;

          then

          reconsider f1 = <*(f . (i + 1))*> as FinSequence of S by FINSEQ_1: 36;

          

           A11: ( Union f1) = ( union ( rng f1)) by CARD_3:def 4

          .= ( union {(f . (i + 1))}) by FINSEQ_1: 38

          .= (f . (i + 1)) by ZFMISC_1: 25;

          ( Union f) = (( Union (f | i)) \/ ( Union f1)) by A10, ROUGHS_1: 5;

          

          then

           A13: (G \ ( Union f)) = ((G \ ( Union (f | i))) \ (f . (i + 1))) by A11, XBOOLE_1: 41

          .= (( Union h) \ (f . (i + 1))) by A12;

          deffunc F( Nat) = ((h . $1) \ (f . (i + 1)));

          consider V be FinSequence such that

           A14: ( len V) = ( len h) & for k be Nat st k in ( dom V) holds (V . k) = F(k) from FINSEQ_1:sch 2;

          

           A19: for k be Nat st k in ( dom V) holds ex D be disjoint_valued FinSequence of S st (V . k) = ( Union D)

          proof

            let k be Nat;

            assume

             A15: k in ( dom V);

            k in ( dom h) by A14, A15, FINSEQ_3: 29;

            then

             A16: (h . k) in ( rng h) by FUNCT_1: 3;

            (i + 1) in ( Seg ( len f)) by A9, FINSEQ_1: 4;

            then (i + 1) in ( dom f) by FINSEQ_1:def 3;

            then (f . (i + 1)) in ( rng f) by FUNCT_1: 3;

            then

            consider D be disjoint_valued FinSequence of S such that

             A18: ((h . k) \ (f . (i + 1))) = ( Union D) by A16, SRINGS_3:def 1;

            take D;

            thus (V . k) = ( Union D) by A15, A14, A18;

          end;

          defpred P[ Nat, object] means ex D be disjoint_valued FinSequence of S st $2 = D & (V . $1) = ( Union D);

          

           P1: for k be Nat st k in ( Seg ( len V)) holds ex x be object st P[k, x]

          proof

            let k be Nat;

            assume k in ( Seg ( len V));

            then k in ( dom V) by FINSEQ_1:def 3;

            then

            consider D be disjoint_valued FinSequence of S such that

             P2: (V . k) = ( Union D) by A19;

            take D;

            thus thesis by P2;

          end;

          consider FinS be FinSequence such that

           P3: ( dom FinS) = ( Seg ( len V)) & for k be Nat st k in ( Seg ( len V)) holds P[k, (FinS . k)] from FINSEQ_1:sch 1( P1);

          now

            let a be object;

            assume a in ( rng FinS);

            then

            consider x be object such that

             P4: x in ( dom FinS) & a = (FinS . x) by FUNCT_1:def 3;

            consider D be disjoint_valued FinSequence of S such that

             P5: (FinS . x) = D & (V . x) = ( Union D) by P3, P4;

            thus a is FinSequence of S by P4, P5;

          end;

          then

          reconsider Y = ( rng FinS) as FinSequenceSet of S by FINSEQ_2:def 3;

          reconsider FinS as FinSequence of Y by FINSEQ_1:def 4;

          

           H1: for n,m be Nat st n <> m holds ( union ( rng (FinS . n))) misses ( union ( rng (FinS . m)))

          proof

            let n,m be Nat;

            assume

             H2: n <> m;

            per cases ;

              suppose

               H3: n in ( dom FinS) & m in ( dom FinS);

              then

              consider D1 be disjoint_valued FinSequence of S such that

               H4: (FinS . n) = D1 & (V . n) = ( Union D1) by P3;

              consider D2 be disjoint_valued FinSequence of S such that

               H5: (FinS . m) = D2 & (V . m) = ( Union D2) by H3, P3;

              

               H6: (V . n) = ( union ( rng (FinS . n))) & (V . m) = ( union ( rng (FinS . m))) by H4, H5, CARD_3:def 4;

              n in ( dom V) & m in ( dom V) by H3, P3, FINSEQ_1:def 3;

              then

               P15: (V . n) = ((h . n) \ (f . (i + 1))) & (V . m) = ((h . m) \ (f . (i + 1))) by A14;

              then (V . n) misses (h . m) by XBOOLE_1: 80, H2, PROB_2:def 2;

              hence ( union ( rng (FinS . n))) misses ( union ( rng (FinS . m))) by H6, P15, XBOOLE_1: 80;

            end;

              suppose not n in ( dom FinS) or not m in ( dom FinS);

              then (FinS . n) = {} or (FinS . m) = {} by FUNCT_1:def 2;

              then ( rng (FinS . n)) = {} or ( rng (FinS . m)) = {} ;

              hence ( union ( rng (FinS . n))) misses ( union ( rng (FinS . m))) by ZFMISC_1: 2, XBOOLE_1: 65;

            end;

          end;

          for n be Nat holds (FinS . n) is disjoint_valued

          proof

            let n be Nat;

            per cases ;

              suppose not n in ( dom FinS);

              hence (FinS . n) is disjoint_valued by FUNCT_1:def 2;

            end;

              suppose n in ( dom FinS);

              then ex D be disjoint_valued FinSequence of S st (FinS . n) = D & (V . n) = ( Union D) by P3;

              hence (FinS . n) is disjoint_valued;

            end;

          end;

          then

          reconsider H = ( joined_FinSeq FinS) as disjoint_valued FinSequence of S by H1, Th14;

          take H;

          ( Union H) = ( union ( rng H)) by CARD_3:def 4;

          then

           X1: ( Union H) = ( union ( union { ( rng (FinS . n)) where n be Nat : n in ( dom FinS) })) by Th15;

          

           X2: (G \ ( Union f)) = (( union ( rng h)) \ (f . (i + 1))) by CARD_3:def 4, A13;

          now

            let x be object;

            assume

             B0: x in (( union ( rng h)) \ (f . (i + 1)));

            then

            consider A be set such that

             B2: x in A & A in ( rng h) by TARSKI:def 4;

            consider k be object such that

             B3: k in ( dom h) & A = (h . k) by B2, FUNCT_1:def 3;

            reconsider k as Nat by B3;

            

             B4: k in ( dom V) by A14, B3, FINSEQ_3: 29;

            

             B5: k in ( dom FinS) by P3, FINSEQ_1:def 3, A14, B3;

            then

            consider D be disjoint_valued FinSequence of S such that

             B6: (FinS . k) = D & (V . k) = ( Union D) by P3;

            

             B7: (V . k) = ( union ( rng (FinS . k))) by B6, CARD_3:def 4;

            x in ( union ( rng h)) & not x in (f . (i + 1)) by B0, XBOOLE_0:def 5;

            then x in ((h . k) \ (f . (i + 1))) by B2, B3, XBOOLE_0:def 5;

            then x in (V . k) by A14, B4;

            then

            consider V be set such that

             B8: x in V & V in ( rng (FinS . k)) by B7, TARSKI:def 4;

            ( rng (FinS . k)) in { ( rng (FinS . n)) where n be Nat : n in ( dom FinS) } by B5;

            then V in ( union { ( rng (FinS . n)) where n be Nat : n in ( dom FinS) }) by B8, TARSKI:def 4;

            hence x in ( union ( union { ( rng (FinS . n)) where n be Nat : n in ( dom FinS) })) by B8, TARSKI:def 4;

          end;

          then

           B9: (G \ ( Union f)) c= ( Union H) by X1, X2, TARSKI:def 3;

          now

            let x be object;

            assume x in ( union ( union { ( rng (FinS . n)) where n be Nat : n in ( dom FinS) }));

            then

            consider A be set such that

             C1: x in A & A in ( union { ( rng (FinS . n)) where n be Nat : n in ( dom FinS) }) by TARSKI:def 4;

            consider D1 be set such that

             C2: A in D1 & D1 in { ( rng (FinS . n)) where n be Nat : n in ( dom FinS) } by C1, TARSKI:def 4;

            consider k be Nat such that

             C3: D1 = ( rng (FinS . k)) & k in ( dom FinS) by C2;

            consider D2 be disjoint_valued FinSequence of S such that

             C4: (FinS . k) = D2 & (V . k) = ( Union D2) by C3, P3;

            

             C5: k in ( dom V) by C3, P3, FINSEQ_1:def 3;

            then (V . k) = ((h . k) \ (f . (i + 1))) by A14;

            then ((h . k) \ (f . (i + 1))) = ( union D1) by C3, C4, CARD_3:def 4;

            then

             C6: x in ((h . k) \ (f . (i + 1))) by C1, C2, TARSKI:def 4;

            then

             C7: x in (h . k) & not x in (f . (i + 1)) by XBOOLE_0:def 5;

            ( dom V) = ( dom h) by A14, FINSEQ_3: 29;

            then (h . k) in ( rng h) by C5, FUNCT_1: 3;

            then x in ( union ( rng h)) by C6, TARSKI:def 4;

            hence x in (( union ( rng h)) \ (f . (i + 1))) by C7, XBOOLE_0:def 5;

          end;

          then ( Union H) c= (G \ ( Union f)) by X1, X2, TARSKI:def 3;

          hence (G \ ( Union f)) = ( Union H) by B9, XBOOLE_0:def 10;

        end;

        hence P[(i + 1)];

      end;

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

      then for f be FinSequence of S st ( len f) = ( len F) holds ex H be disjoint_valued FinSequence of S st (G \ ( Union f)) = ( Union H);

      hence thesis;

    end;

    registration

      let X be set, P be with_empty_element semi-diff-closed cap-closed Subset-Family of X;

      cluster disjoint_valued for sequence of P;

      existence

      proof

        consider F be sequence of { {} } such that

         A1: for n be Element of NAT holds (F . n) = {} by MEASURE1: 16;

         { {} } c= P by ZFMISC_1: 31, SETFAM_1:def 8;

        then

        reconsider F as sequence of P by FUNCT_2: 7;

        take F;

        for x,y be object st x <> y holds (F . x) misses (F . y)

        proof

          let x,y be object;

          assume x <> y;

          per cases ;

            suppose x in ( dom F);

            then (F . x) = {} by A1;

            hence (F . x) misses (F . y) by XBOOLE_1: 65;

          end;

            suppose not x in ( dom F);

            then (F . x) = {} by FUNCT_1:def 2;

            hence (F . x) misses (F . y) by XBOOLE_1: 65;

          end;

        end;

        hence F is disjoint_valued by PROB_2:def 2;

      end;

    end

    

     LM: for X be set, P be non empty Subset-Family of X holds (P --> 0. ) is nonnegative & (P --> 0. ) is additive & (P --> 0. ) is zeroed

    proof

      let X be set, P be non empty Subset-Family of X;

      set M = (P --> 0. );

      for A be Element of P holds 0. <= (M . A);

      hence (P --> 0. ) is nonnegative by MEASURE1:def 2;

      now

        let A,B be Element of P;

        assume A misses B & (A \/ B) in P;

        then

        reconsider D = (A \/ B) as Element of P;

        (M . A) = 0. & (M . B) = 0. & (M . D) = 0. by FUNCOP_1: 7;

        hence (M . (A \/ B)) = ((M . A) + (M . B));

      end;

      hence (P --> 0. ) is additive by MEASURE1:def 3;

      per cases ;

        suppose {} in P;

        then ((P --> 0. ) . {} ) = 0. by FUNCOP_1: 7;

        hence (P --> 0. ) is zeroed by VALUED_0:def 19;

      end;

        suppose not {} in P;

        then not {} in ( dom (P --> 0. ));

        then ((P --> 0. ) . {} ) = 0 by FUNCT_1:def 2;

        hence (P --> 0. ) is zeroed by VALUED_0:def 19;

      end;

    end;

    registration

      let X be set, P be non empty Subset-Family of X;

      cluster nonnegative additive zeroed for Function of P, ExtREAL ;

      existence

      proof

        reconsider M = (P --> 0. ) as Function of P, ExtREAL ;

        take M;

        thus thesis by LM;

      end;

    end

    registration

      let X be set, P be with_empty_element Subset-Family of X;

      cluster disjoint_valued for Function of NAT , P;

      existence

      proof

         {} in P by SETFAM_1:def 8;

        then

        reconsider F = ( NAT --> {} ) as Function of NAT , P by FUNCOP_1: 46;

        take F;

        now

          let i,j be object;

          assume i <> j;

          per cases ;

            suppose i in ( dom F);

            thus (F . i) misses (F . j) by XBOOLE_1: 65;

          end;

            suppose not i in ( dom F);

            thus (F . i) misses (F . j) by XBOOLE_1: 65;

          end;

        end;

        hence F is disjoint_valued by PROB_2:def 2;

      end;

    end

    definition

      let X be set, P be with_empty_element Subset-Family of X;

      :: MEASURE9:def7

      mode pre-Measure of P -> nonnegative zeroed Function of P, ExtREAL means

      : Def8: (for F be disjoint_valued FinSequence of P st ( Union F) in P holds (it . ( Union F)) = ( Sum (it * F))) & (for K be disjoint_valued Function of NAT , P st ( Union K) in P holds (it . ( Union K)) <= ( SUM (it * K)));

      existence

      proof

        reconsider M = (P --> 0. ) as Function of P, ExtREAL ;

        (for x be Element of P holds 0. <= (M . x)) & (M . {} ) = 0 by FUNCOP_1: 7, SETFAM_1:def 8;

        then

        reconsider M as nonnegative zeroed Function of P, ExtREAL by MEASURE1:def 2, VALUED_0:def 19;

        take M;

         0 is Element of REAL by XREAL_0:def 1;

        then

        reconsider m = (P --> 0 ) as Function of P, REAL by FUNCOP_1: 46;

        

         A2: for F be disjoint_valued FinSequence of P st ( Union F) in P holds (M . ( Union F)) = ( Sum (M * F))

        proof

          let F be disjoint_valued FinSequence of P;

          assume ( Union F) in P;

          then

           A3: (M . ( Union F)) = 0. by FUNCOP_1: 7;

          ( rng F) c= P & ( dom M) = P & ( dom m) = P by FUNCT_2:def 1;

          then

           A4: ( dom (M * F)) = ( dom F) & ( dom (m * F)) = ( dom F) by RELAT_1: 27;

          

           A7: ( Sum (M * F)) = ( Sum (m * F)) by MESFUNC3: 2;

          

           A8: (m * F) = ((F " P) --> 0 ) by FUNCOP_1: 19;

          then ( dom F) = (F " P) by A4, FUNCOP_1: 13;

          then ( Seg ( len F)) = (F " P) by FINSEQ_1:def 3;

          then (m * F) = (( len F) |-> 0 ) by A8, FINSEQ_2:def 2;

          hence (M . ( Union F)) = ( Sum (M * F)) by A3, A7, RVSUM_1: 81;

        end;

        for K be disjoint_valued Function of NAT , P st ( Union K) in P holds (M . ( Union K)) <= ( SUM (M * K))

        proof

          let K be disjoint_valued Function of NAT , P;

          assume ( Union K) in P;

          then

           A10: (M . ( Union K)) = 0. by FUNCOP_1: 7;

          now

            let n be Element of NAT ;

            ((M * K) . n) = (M . (K . n)) by FUNCT_2: 15;

            hence ((M * K) . n) = 0. by FUNCOP_1: 7;

          end;

          hence (M . ( Union K)) <= ( SUM (M * K)) by A10, MEASURE7: 1;

        end;

        hence thesis by A2;

      end;

    end

    theorem :: MEASURE9:32

    for X be with_empty_element set, F be FinSequence of X holds ex G be Function of NAT , X st (for i be Nat holds (F . i) = (G . i)) & ( Union F) = ( Union G)

    proof

      let X be with_empty_element set;

      let F be FinSequence of X;

      defpred P[ Element of NAT , set] means ($1 in ( dom F) implies (F . $1) = $2) & ( not $1 in ( dom F) implies $2 = {} );

      

       A1: for i be Element of NAT holds ex y be Element of X st P[i, y]

      proof

        let i be Element of NAT ;

        per cases ;

          suppose

           A2: i in ( dom F);

          then (F . i) in ( rng F) & ( rng F) c= X by FUNCT_1: 3;

          then

          reconsider y = (F . i) as Element of X;

          take y;

          thus P[i, y] by A2;

        end;

          suppose

           A3: not i in ( dom F);

          reconsider y = {} as Element of X by SETFAM_1:def 8;

          take y;

          thus P[i, y] by A3;

        end;

      end;

      consider G be Function of NAT , X such that

       A4: for i be Element of NAT holds P[i, (G . i)] from FUNCT_2:sch 3( A1);

      take G;

       A5:

      now

        let i be Nat;

        per cases ;

          suppose i in ( dom F);

          hence (F . i) = (G . i) by A4;

        end;

          suppose not i in ( dom F);

          reconsider j = i as Element of NAT by ORDINAL1:def 12;

           P[j, (G . j)] by A4;

          hence (F . i) = (G . i) by FUNCT_1:def 2;

        end;

      end;

      

       B1: ( Union F) = ( union ( rng F)) & ( Union G) = ( union ( rng G)) by CARD_3:def 4;

      now

        let x be object;

        assume x in ( Union F);

        then

        consider A be set such that

         A7: x in A & A in ( rng F) by B1, TARSKI:def 4;

        consider k be object such that

         A8: k in ( dom F) & A = (F . k) by A7, FUNCT_1:def 3;

        reconsider k as Nat by A8;

        ( dom G) = NAT by FUNCT_2:def 1;

        then A = (G . k) & (G . k) in ( rng G) by A5, A8, FUNCT_1: 3;

        hence x in ( Union G) by A7, B1, TARSKI:def 4;

      end;

      then

       A9: ( Union F) c= ( Union G) by TARSKI:def 3;

      now

        let x be object;

        assume x in ( Union G);

        then

        consider A be set such that

         A10: x in A & A in ( rng G) by B1, TARSKI:def 4;

        consider k be object such that

         A11: k in ( dom G) & A = (G . k) by A10, FUNCT_1:def 3;

        reconsider k as Nat by A11;

         A12:

        now

          assume not k in ( dom F);

          then (F . k) = {} by FUNCT_1:def 2;

          hence contradiction by A5, A10, A11;

        end;

        

         A13: (F . k) = (G . k) by A5;

        (F . k) in ( rng F) by A12, FUNCT_1: 3;

        hence x in ( Union F) by B1, A10, A11, A13, TARSKI:def 4;

      end;

      then ( Union G) c= ( Union F) by TARSKI:def 3;

      hence thesis by A5, A9, XBOOLE_0:def 10;

    end;

    theorem :: MEASURE9:33

    for X be non empty set, F be FinSequence of X, G be Function of NAT , X st (for i be Nat holds (F . i) = (G . i)) holds F is disjoint_valued iff G is disjoint_valued

    proof

      let X be non empty set, F be FinSequence of X, G be Function of NAT , X;

      assume

       A1: for i be Nat holds (F . i) = (G . i);

      hereby

        assume

         A2: F is disjoint_valued;

        now

          let x,y be object;

          assume

           A3: x <> y;

          per cases ;

            suppose x in ( dom F) & y in ( dom F);

            then (G . x) = (F . x) & (G . y) = (F . y) by A1;

            hence (G . x) misses (G . y) by A2, A3, PROB_2:def 2;

          end;

            suppose not x in ( dom F) & x in ( dom G);

            then (F . x) = {} & (G . x) = (F . x) by A1, FUNCT_1:def 2;

            hence (G . x) misses (G . y) by XBOOLE_1: 65;

          end;

            suppose not x in ( dom F) & not x in ( dom G);

            then (G . x) = {} by FUNCT_1:def 2;

            hence (G . x) misses (G . y) by XBOOLE_1: 65;

          end;

            suppose not y in ( dom F) & y in ( dom G);

            then (F . y) = {} & (G . y) = (F . y) by A1, FUNCT_1:def 2;

            hence (G . x) misses (G . y) by XBOOLE_1: 65;

          end;

            suppose not y in ( dom F) & not y in ( dom G);

            then (G . y) = {} by FUNCT_1:def 2;

            hence (G . x) misses (G . y) by XBOOLE_1: 65;

          end;

        end;

        hence G is disjoint_valued by PROB_2:def 2;

      end;

      assume

       A8: G is disjoint_valued;

      now

        let x,y be object;

        assume

         A9: x <> y;

        per cases ;

          suppose x in ( dom G) & y in ( dom G);

          then (F . x) = (G . x) & (F . y) = (G . y) by A1;

          hence (F . x) misses (F . y) by A8, A9, PROB_2:def 2;

        end;

          suppose

           A10: not x in ( dom G) or not y in ( dom G);

          ( dom F) c= NAT ;

          then ( dom F) c= ( dom G) by FUNCT_2:def 1;

          then not x in ( dom F) or not y in ( dom F) by A10;

          then (F . x) = {} or (F . y) = {} by FUNCT_1:def 2;

          hence (F . x) misses (F . y) by XBOOLE_1: 65;

        end;

      end;

      hence F is disjoint_valued by PROB_2:def 2;

    end;

    theorem :: MEASURE9:34

    for F be FinSequence of ExtREAL , G be ExtREAL_sequence st (for i be Nat holds (F . i) = (G . i)) holds F is nonnegative iff G is nonnegative

    proof

      let F be FinSequence of ExtREAL , G be ExtREAL_sequence;

      assume

       A1: for i be Nat holds (F . i) = (G . i);

      hereby

        assume

         A3: F is nonnegative;

        now

          let i be object;

          assume

           A4: i in ( dom G);

          per cases ;

            suppose i in ( dom F);

            then (G . i) = (F . i) by A1;

            hence (G . i) >= 0 by A3, SUPINF_2: 51;

          end;

            suppose not i in ( dom F);

            then (F . i) = 0 by FUNCT_1:def 2;

            hence (G . i) >= 0 by A1, A4;

          end;

        end;

        hence G is nonnegative by SUPINF_2: 52;

      end;

      assume

       A5: G is nonnegative;

      now

        let i be object;

        per cases ;

          suppose i in ( dom F);

          then (F . i) = (G . i) by A1;

          hence (F . i) >= 0 by A5, SUPINF_2: 51;

        end;

          suppose not i in ( dom F);

          hence (F . i) >= 0 by FUNCT_1:def 2;

        end;

      end;

      hence F is nonnegative by SUPINF_2: 51;

    end;

    

     LL1: <* +infty *> is nonnegative & <* +infty *> is without-infty

    proof

      set F = <* +infty *>;

      now

        let i be object;

        per cases ;

          suppose i in ( dom F);

          then i in ( Seg 1) by FINSEQ_1: 38;

          then i = 1 by TARSKI:def 1, FINSEQ_1: 2;

          hence (F . i) >= 0 by FINSEQ_1: 40;

        end;

          suppose not i in ( dom F);

          hence (F . i) >= 0 by FUNCT_1:def 2;

        end;

      end;

      hence F is nonnegative by SUPINF_2: 51;

      hence F is without-infty;

    end;

    

     LL2: <* -infty *> is nonpositive & <* -infty *> is without+infty

    proof

      set F = <* -infty *>;

      now

        let i be object;

        per cases ;

          suppose i in ( dom F);

          then i in ( Seg 1) by FINSEQ_1: 38;

          then i = 1 by TARSKI:def 1, FINSEQ_1: 2;

          hence (F . i) <= 0 by FINSEQ_1: 40;

        end;

          suppose not i in ( dom F);

          hence (F . i) <= 0 by FUNCT_1:def 2;

        end;

      end;

      hence F is nonpositive by MESFUNC5: 8;

      hence F is without+infty;

    end;

    registration

      cluster nonnegative for FinSequence of ExtREAL ;

      existence by LL1;

      cluster without-infty for FinSequence of ExtREAL ;

      existence by LL1;

      cluster nonpositive for FinSequence of ExtREAL ;

      existence by LL2;

      cluster without+infty for FinSequence of ExtREAL ;

      existence by LL2;

      cluster nonnegative -> without-infty for FinSequence of ExtREAL ;

      correctness ;

      cluster nonpositive -> without+infty for FinSequence of ExtREAL ;

      correctness ;

    end

    registration

      let X,Y be non empty set, F be without-infty Function of Y, ExtREAL , G be Function of X, Y;

      cluster (F * G) -> without-infty;

      correctness

      proof

        for x be object holds -infty < ((F * G) . x)

        proof

          let x be object;

          per cases ;

            suppose x in ( dom (F * G));

            then ((F * G) . x) = (F . (G . x)) by FUNCT_1: 12;

            hence -infty < ((F * G) . x) by MESFUNC5:def 5;

          end;

            suppose not x in ( dom (F * G));

            hence -infty < ((F * G) . x) by FUNCT_1:def 2;

          end;

        end;

        hence thesis by MESFUNC5:def 5;

      end;

    end

    registration

      let X,Y be non empty set, F be nonnegative Function of Y, ExtREAL , G be Function of X, Y;

      cluster (F * G) -> nonnegative;

      correctness by MEASURE1: 25;

    end

    theorem :: MEASURE9:35

    

     Th33: for a be R_eal holds ( Sum <*a*>) = a

    proof

      let a be R_eal;

      consider f be sequence of ExtREAL such that

       A1: ( Sum <*a*>) = (f . ( len <*a*>)) & (f . 0 ) = 0. & for i be Nat st i < ( len <*a*>) holds (f . (i + 1)) = ((f . i) + ( <*a*> . (i + 1))) by EXTREAL1:def 2;

      

       A2: ( len <*a*>) = 1 by FINSEQ_1: 39;

      (f . ( 0 + 1)) = ((f . 0 ) + ( <*a*> . ( 0 + 1))) by A1

      .= ( 0 + a) by A1, FINSEQ_1: 40;

      hence ( Sum <*a*>) = a by A1, A2, XXREAL_3: 4;

    end;

    theorem :: MEASURE9:36

    

     Th34: for F be FinSequence of ExtREAL , k be Nat holds (F is without-infty implies (F | k) is without-infty) & (F is without+infty implies (F | k) is without+infty)

    proof

      let F be FinSequence of ExtREAL , k be Nat;

      hereby

        assume

         A1: F is without-infty;

        now

          assume -infty in ( rng (F | k));

          then

          consider i be Element of NAT such that

           A2: i in ( dom (F | k)) & -infty = ((F | k) . i) by PARTFUN1: 3;

          ( dom (F | k)) c= ( dom F) by RELAT_1: 60;

          then i in ( dom F) & ((F | k) . i) = (F . i) by A2, FUNCT_1: 47;

          then -infty in ( rng F) by A2, FUNCT_1: 3;

          hence contradiction by A1, MESFUNC5:def 3;

        end;

        hence (F | k) is without-infty by MESFUNC5:def 3;

      end;

      assume

       A3: F is without+infty;

      now

        assume +infty in ( rng (F | k));

        then

        consider i be Element of NAT such that

         A4: i in ( dom (F | k)) & +infty = ((F | k) . i) by PARTFUN1: 3;

        ( dom (F | k)) c= ( dom F) by RELAT_1: 60;

        then i in ( dom F) & ((F | k) . i) = (F . i) by A4, FUNCT_1: 47;

        then +infty in ( rng F) by A4, FUNCT_1: 3;

        hence contradiction by A3, MESFUNC5:def 4;

      end;

      hence (F | k) is without+infty by MESFUNC5:def 4;

    end;

    theorem :: MEASURE9:37

    

     Th35: for F be without-infty FinSequence of ExtREAL , G be ExtREAL_sequence st (for i be Nat holds (F . i) = (G . i)) holds for i be Nat holds ( Sum (F | i)) = (( Partial_Sums G) . i)

    proof

      let F be without-infty FinSequence of ExtREAL , G be ExtREAL_sequence;

      assume

       A1: for i be Nat holds (F . i) = (G . i);

      hereby

        let i be Nat;

        defpred P[ Nat] means ( Sum (F | $1)) = (( Partial_Sums G) . $1);

        

         A3: ex f0 be sequence of ExtREAL st ( Sum (F | 0 )) = (f0 . ( len (F | 0 ))) & (f0 . 0 ) = 0. & for i be Nat st i < ( len (F | 0 )) holds (f0 . (i + 1)) = ((f0 . i) + ((F | 0 ) . (i + 1))) by EXTREAL1:def 2;

         not 0 in ( Seg ( len F)) by FINSEQ_1: 1;

        then not 0 in ( dom F) by FINSEQ_1:def 3;

        then (F . 0 ) = 0 by FUNCT_1:def 2;

        then (G . 0 ) = 0 by A1;

        then

         A4: P[ 0 ] by A3, MESFUNC9:def 1;

        

         A5: for k be Nat st P[k] holds P[(k + 1)]

        proof

          let k be Nat;

          assume

           A6: P[k];

          (F | k) is without-infty by Th34;

          then

           A7: not -infty in ( rng (F | k)) by MESFUNC5:def 3;

           A8:

          now

            assume -infty in ( rng <*(F . (k + 1))*>);

            then -infty in {(F . (k + 1))} by FINSEQ_1: 39;

            then

             A9: (F . (k + 1)) = -infty by TARSKI:def 1;

            per cases ;

              suppose (k + 1) in ( dom F);

              then (F . (k + 1)) in ( rng F) by FUNCT_1: 3;

              hence contradiction by A9, MESFUNC5:def 3;

            end;

              suppose not (k + 1) in ( dom F);

              hence contradiction by A9, FUNCT_1:def 2;

            end;

          end;

          per cases ;

            suppose (k + 1) <= ( len F);

            then (F | (k + 1)) = ((F | k) ^ <*(F . (k + 1))*>) by NAT_1: 13, FINSEQ_5: 83;

            

            then ( Sum (F | (k + 1))) = (( Sum (F | k)) + ( Sum <*(F . (k + 1))*>)) by A7, A8, EXTREAL1: 10

            .= ((( Partial_Sums G) . k) + (F . (k + 1))) by A6, Th33

            .= ((( Partial_Sums G) . k) + (G . (k + 1))) by A1;

            hence P[(k + 1)] by MESFUNC9:def 1;

          end;

            suppose

             A10: (k + 1) > ( len F);

            then

             A11: (F | k) = F & (F | (k + 1)) = F by NAT_1: 13, FINSEQ_1: 58;

             not (k + 1) in ( dom F) by A10, FINSEQ_3: 25;

            then (F . (k + 1)) = 0 by FUNCT_1:def 2;

            then (G . (k + 1)) = 0 by A1;

            then (( Partial_Sums G) . (k + 1)) = ((( Partial_Sums G) . k) + 0 ) by MESFUNC9:def 1;

            hence P[(k + 1)] by A6, A11, XXREAL_3: 4;

          end;

        end;

        for k be Nat holds P[k] from NAT_1:sch 2( A4, A5);

        hence ( Sum (F | i)) = (( Partial_Sums G) . i);

      end;

    end;

    theorem :: MEASURE9:38

    for F be without-infty FinSequence of ExtREAL , G be ExtREAL_sequence st (for i be Nat holds (F . i) = (G . i)) holds G is summable & ( Sum F) = ( Sum G)

    proof

      let F be without-infty FinSequence of ExtREAL , G be ExtREAL_sequence;

      assume

       A1: for i be Nat holds (F . i) = (G . i);

      then

       A2: ( Sum (F | ( len F))) = (( Partial_Sums G) . ( len F)) by Th35;

      defpred P[ Nat] means ( Sum F) = (( Partial_Sums G) . (( len F) + $1));

      

       B1: P[ 0 ] by A2, FINSEQ_1: 58;

      

       B2: for k be Nat st P[k] holds P[(k + 1)]

      proof

        let k be Nat;

        assume

         A3: P[k];

        ( len F) < (( len F) + (k + 1)) by NAT_1: 11, NAT_1: 19;

        then not ((( len F) + k) + 1) in ( dom F) by FINSEQ_3: 25;

        then (F . ((( len F) + k) + 1)) = 0 by FUNCT_1:def 2;

        then

         A4: (G . ((( len F) + k) + 1)) = 0 by A1;

        (( Partial_Sums G) . (( len F) + (k + 1))) = ((( Partial_Sums G) . (( len F) + k)) + (G . ((( len F) + k) + 1))) by MESFUNC9:def 1

        .= (( Partial_Sums G) . (( len F) + k)) by A4, XXREAL_3: 4;

        hence P[(k + 1)] by A3;

      end;

      

       A5: for k be Nat holds P[k] from NAT_1:sch 2( B1, B2);

      hereby

        per cases by XXREAL_0: 14;

          suppose ( Sum F) in REAL ;

          then

          reconsider r = ( Sum F) as Real;

          

           B1: for p be Real st 0 < p holds ex n be Nat st for m be Nat st n <= m holds |.((( Partial_Sums G) . m) - r) qua ExtReal.| < p

          proof

            let p be Real;

            assume

             A6: 0 < p;

            take n = ( len F);

            now

              let m be Nat;

              assume ( len F) <= m;

              then

              reconsider k = (m - n) as Nat by NAT_1: 21;

              m = (n + k);

              then (( Partial_Sums G) . m) = ( Sum F) by A5;

              hence |.((( Partial_Sums G) . m) - r) qua ExtReal.| < p by A6, XXREAL_3: 7, EXTREAL1: 16;

            end;

            hence thesis;

          end;

          then

           B2: ( Partial_Sums G) is convergent_to_finite_number by MESFUNC5:def 8;

          hence G is summable by MESFUNC9:def 2;

          ( lim ( Partial_Sums G)) = ( Sum F) by B1, B2, MESFUNC5:def 12;

          hence ( Sum F) = ( Sum G) by MESFUNC9:def 3;

        end;

          suppose

           A7: ( Sum F) = +infty ;

          now

            let g be Real;

            assume 0 < g;

            thus ex n be Nat st for m be Nat st n <= m holds g <= (( Partial_Sums G) . m)

            proof

              take n = ( len F);

              hereby

                let m be Nat;

                assume n <= m;

                then

                reconsider k = (m - n) as Nat by NAT_1: 21;

                m = (n + k);

                then (( Partial_Sums G) . m) = +infty by A5, A7;

                hence g <= (( Partial_Sums G) . m) by XXREAL_0: 3;

              end;

            end;

          end;

          then

           B5: ( Partial_Sums G) is convergent_to_+infty by MESFUNC5:def 9;

          hence G is summable by MESFUNC9:def 2;

          ( lim ( Partial_Sums G)) = ( Sum F) by A7, B5, MESFUNC5:def 12;

          hence ( Sum F) = ( Sum G) by MESFUNC9:def 3;

        end;

          suppose

           A8: ( Sum F) = -infty ;

          now

            let g be Real;

            assume g < 0 ;

            thus ex n be Nat st for m be Nat st n <= m holds (( Partial_Sums G) . m) <= g

            proof

              take n = ( len F);

              hereby

                let m be Nat;

                assume n <= m;

                then

                reconsider k = (m - n) as Nat by NAT_1: 21;

                m = (n + k);

                then (( Partial_Sums G) . m) = -infty by A5, A8;

                hence (( Partial_Sums G) . m) <= g by XXREAL_0: 5;

              end;

            end;

          end;

          then

           B8: ( Partial_Sums G) is convergent_to_-infty by MESFUNC5:def 10;

          hence G is summable by MESFUNC9:def 2;

          ( lim ( Partial_Sums G)) = ( Sum F) by A8, B8, MESFUNC5:def 12;

          hence ( Sum F) = ( Sum G) by MESFUNC9:def 3;

        end;

      end;

    end;

    theorem :: MEASURE9:39

    for X be set, S be with_empty_element semi-diff-closed cap-closed Subset-Family of X, F be disjoint_valued FinSequence of S, R be non empty preBoolean Subset-Family of X st S c= R & ( Union F) in R holds for i be Nat holds ( Union (F | i)) in R

    proof

      let X be set, S be with_empty_element semi-diff-closed cap-closed Subset-Family of X, F be disjoint_valued FinSequence of S, R be non empty preBoolean Subset-Family of X;

      assume

       A1: S c= R & ( Union F) in R;

      defpred P[ Nat] means ( Union (F | $1)) in R;

      ( union ( rng (F | 0 ))) = {} by ZFMISC_1: 2;

      then ( Union (F | 0 )) = {} by CARD_3:def 4;

      then

       A3: P[ 0 ] by FINSUB_1: 7;

      

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

      proof

        let i be Nat;

        assume

         A5: P[i];

        per cases ;

          suppose i >= ( len F);

          then (F | i) = F & (F | (i + 1)) = F by NAT_1: 12, FINSEQ_1: 58;

          hence P[(i + 1)] by A5;

        end;

          suppose i < ( len F);

          then

           A8: (i + 1) <= ( len F) by NAT_1: 13;

          set F1 = (F | (i + 1));

          

           A9: (F1 | i) = (F | i) by NAT_1: 12, FINSEQ_1: 82;

          F1 = ((F1 | i) ^ <*(F1 . (i + 1))*>) by A8, FINSEQ_1: 17, FINSEQ_3: 55;

          then ( rng F1) = (( rng (F1 | i)) \/ ( rng <*(F1 . (i + 1))*>)) by FINSEQ_1: 31;

          then ( rng F1) = (( rng (F | i)) \/ {(F1 . (i + 1))}) by A9, FINSEQ_1: 38;

          then ( rng F1) = (( rng (F | i)) \/ {(F . (i + 1))}) by FINSEQ_3: 112;

          then ( union ( rng F1)) = (( union ( rng (F | i))) \/ ( union {(F . (i + 1))})) by ZFMISC_1: 78;

          then ( Union F1) = (( union ( rng (F | i))) \/ ( union {(F . (i + 1))})) by CARD_3:def 4;

          then ( Union F1) = (( Union (F | i)) \/ ( union {(F . (i + 1))})) by CARD_3:def 4;

          then

           A11: ( Union F1) = (( Union (F | i)) \/ (F . (i + 1))) by ZFMISC_1: 25;

          (i + 1) in ( dom F) by A8, NAT_1: 12, FINSEQ_3: 25;

          then (F . (i + 1)) in ( rng F) by FUNCT_1: 3;

          then (F . (i + 1)) in S;

          hence P[(i + 1)] by A1, A5, A11, FINSUB_1:def 1;

        end;

      end;

      for i be Nat holds P[i] from NAT_1:sch 2( A3, A4);

      hence thesis;

    end;

    theorem :: MEASURE9:40

    for X be set, S be with_empty_element semi-diff-closed cap-closed Subset-Family of X, P be pre-Measure of S, F1,F2 be disjoint_valued FinSequence of S st ( Union F1) in S & ( Union F1) = ( Union F2) holds (P . ( Union F1)) = (P . ( Union F2));

    theorem :: MEASURE9:41

    

     FStoMAT1: for S be non empty cap-closed set, F1,F2 be FinSequence of S holds ex Mx be Matrix of ( len F1), ( len F2), S st for i,j be Nat st [i, j] in ( Indices Mx) holds (Mx * (i,j)) = ((F1 . i) /\ (F2 . j))

    proof

      let S be non empty cap-closed set;

      let F1,F2 be FinSequence of S;

      defpred P[ Nat, Nat, set] means $3 = ((F1 . $1) /\ (F2 . $2));

      

       A2: for i,j be Nat st [i, j] in [:( Seg ( len F1)), ( Seg ( len F2)):] holds ex K be Element of S st P[i, j, K]

      proof

        let i,j be Nat;

        assume [i, j] in [:( Seg ( len F1)), ( Seg ( len F2)):];

        then i in ( Seg ( len F1)) & j in ( Seg ( len F2)) by ZFMISC_1: 87;

        then i in ( dom F1) & j in ( dom F2) by FINSEQ_1:def 3;

        then (F1 . i) in ( rng F1) & (F2 . j) in ( rng F2) by FUNCT_1: 3;

        then ((F1 . i) /\ (F2 . j)) in S by FINSUB_1:def 2;

        hence thesis;

      end;

      consider Mx be Matrix of ( len F1), ( len F2), S such that

       A3: for i,j be Nat st [i, j] in ( Indices Mx) holds P[i, j, (Mx * (i,j))] from MATRIX_0:sch 2( A2);

      take Mx;

      thus thesis by A3;

    end;

    theorem :: MEASURE9:42

    

     Th40: for X be set, S be with_empty_element cap-closed Subset-Family of X, F1,F2 be non empty disjoint_valued FinSequence of S, P be nonnegative zeroed Function of S, ExtREAL , Mx be Matrix of ( len F1), ( len F2), ExtREAL st ( Union F1) = ( Union F2) & (for i,j be Nat st [i, j] in ( Indices Mx) holds (Mx * (i,j)) = (P . ((F1 . i) /\ (F2 . j)))) & (for F be disjoint_valued FinSequence of S st ( Union F) in S holds (P . ( Union F)) = ( Sum (P * F))) holds (for i be Nat st i <= ( len (P * F1)) holds ((P * F1) . i) = (( Sum Mx) . i)) & ( Sum (P * F1)) = ( SumAll Mx)

    proof

      let X be set, S be with_empty_element cap-closed Subset-Family of X, F1,F2 be non empty disjoint_valued FinSequence of S, P be nonnegative zeroed Function of S, ExtREAL , Mx be Matrix of ( len F1), ( len F2), ExtREAL ;

      assume that

       A1: ( Union F1) = ( Union F2) and

       A2: for i,j be Nat st [i, j] in ( Indices Mx) holds (Mx * (i,j)) = (P . ((F1 . i) /\ (F2 . j))) and

       A3: for F be disjoint_valued FinSequence of S st ( Union F) in S holds (P . ( Union F)) = ( Sum (P * F));

      consider Kx be Matrix of ( len F1), ( len F2), S such that

       KX1: for i,j be Nat st [i, j] in ( Indices Kx) holds (Kx * (i,j)) = ((F1 . i) /\ (F2 . j)) by FStoMAT1;

      

       C0: ( len Kx) = ( len F1) & ( len Mx) = ( len F1) by MATRIX_0:def 2;

      then

       C1: ( len (P * F1)) = ( len Mx) & ( len (P * F1)) = ( len Kx) by FINSEQ_2: 33;

      

       C4: ( width Kx) = ( len F2) & ( width Mx) = ( len F2) by C0, MATRIX_0: 20;

      

       C2: ( len (P * F1)) = ( len ( Sum Mx)) by C1, Def5;

      thus

       C6: for i be Nat st i <= ( len (P * F1)) holds ((P * F1) . i) = (( Sum Mx) . i)

      proof

        let i be Nat;

        assume

         E0: i <= ( len (P * F1));

        per cases ;

          suppose i = 0 ;

          then not i in ( dom (P * F1)) & not i in ( dom ( Sum Mx)) by FINSEQ_3: 24;

          then ((P * F1) . i) = 0 & (( Sum Mx) . i) = 0 by FUNCT_1:def 2;

          hence ((P * F1) . i) = (( Sum Mx) . i);

        end;

          suppose i <> 0 ;

          then

           E1: 1 <= i by NAT_1: 14;

          then i in ( dom (P * F1)) by E0, FINSEQ_3: 25;

          then

           E2: i in ( dom F1) by FUNCT_1: 11;

          then (F1 . i) c= ( union ( rng F1)) by FUNCT_1: 3, ZFMISC_1: 74;

          then (F1 . i) c= ( Union F2) by A1, CARD_3:def 4;

          then

           E3: ((F1 . i) /\ ( Union F2)) = (F1 . i) by XBOOLE_1: 28;

          

           E4: (F1 . i) in ( rng F1) by E2, FUNCT_1: 3;

          

           E5: i in ( dom Kx) & i in ( dom Mx) by C1, E0, E1, FINSEQ_3: 25;

          for p,q be object st p <> q holds ((Kx . i) . p) misses ((Kx . i) . q)

          proof

            let p,q be object;

            assume

             SA0: p <> q;

            per cases ;

              suppose

               SA1: p in ( dom (Kx . i)) & q in ( dom (Kx . i));

              then

              reconsider p1 = p, q1 = q as Nat;

              

               E6: [i, p1] in ( Indices Kx) & [i, q1] in ( Indices Kx) by SA1, E5, MATRIX_0: 37;

              (Kx * (i,p1)) = ((Kx . i) . p) & (Kx * (i,q1)) = ((Kx . i) . q) by E6, MATRPROB: 14;

              then ((Kx . i) . p) = ((F1 . i) /\ (F2 . p1)) & ((Kx . i) . q) = ((F1 . i) /\ (F2 . q1)) by E6, KX1;

              hence ((Kx . i) . p) misses ((Kx . i) . q) by SA0, PROB_2:def 2, XBOOLE_1: 76;

            end;

              suppose not p in ( dom (Kx . i));

              then ((Kx . i) . p) = {} by FUNCT_1:def 2;

              hence ((Kx . i) . p) misses ((Kx . i) . q) by XBOOLE_1: 65;

            end;

              suppose not q in ( dom (Kx . i));

              then ((Kx . i) . q) = {} by FUNCT_1:def 2;

              hence ((Kx . i) . p) misses ((Kx . i) . q) by XBOOLE_1: 65;

            end;

          end;

          then

           E8: (Kx . i) is disjoint_valued FinSequence of S by PROB_2:def 2;

          now

            let x be object;

            assume x in ( Union (Kx . i));

            then x in ( union ( rng (Kx . i))) by CARD_3:def 4;

            then

            consider A be set such that

             E9: x in A & A in ( rng (Kx . i)) by TARSKI:def 4;

            consider m be object such that

             E10: m in ( dom (Kx . i)) & A = ((Kx . i) . m) by E9, FUNCT_1:def 3;

            reconsider m as Nat by E10;

            

             E11: [i, m] in ( Indices Kx) by E10, E5, MATRIX_0: 37;

            then ((Kx . i) . m) = (Kx * (i,m)) by MATRPROB: 14;

            then ((Kx . i) . m) = ((F1 . i) /\ (F2 . m)) by E11, KX1;

            then

             E12: x in (F1 . i) & x in (F2 . m) by E9, E10, XBOOLE_0:def 4;

            1 <= m & m <= ( len F2) by E11, MATRIX_0: 33;

            then m in ( dom F2) by FINSEQ_3: 25;

            then (F2 . m) in ( rng F2) by FUNCT_1: 3;

            then x in ( union ( rng F2)) by E12, TARSKI:def 4;

            then x in ( Union F2) by CARD_3:def 4;

            hence x in ((F1 . i) /\ ( Union F2)) by E12, XBOOLE_0:def 4;

          end;

          then

           E13: ( Union (Kx . i)) c= ((F1 . i) /\ ( Union F2)) by TARSKI:def 3;

          now

            let x be object;

            assume x in ((F1 . i) /\ ( Union F2));

            then

             E14: x in (F1 . i) & x in ( Union F2) by XBOOLE_0:def 4;

            then x in ( union ( rng F2)) by CARD_3:def 4;

            then

            consider A be set such that

             E15: x in A & A in ( rng F2) by TARSKI:def 4;

            consider m be object such that

             E16: m in ( dom F2) & A = (F2 . m) by E15, FUNCT_1:def 3;

            reconsider m as Nat by E16;

            1 <= i & i <= ( len F1) & 1 <= m & m <= ( len F2) by E2, E16, FINSEQ_3: 25;

            then

             E17: [i, m] in ( Indices Kx) by MATRIX_0: 31;

            then ((Kx . i) . m) = (Kx * (i,m)) by MATRPROB: 14;

            then ((Kx . i) . m) = ((F1 . i) /\ (F2 . m)) by E17, KX1;

            then

             E18: x in ((Kx . i) . m) by E14, E15, E16, XBOOLE_0:def 4;

            m in ( dom (Kx . i)) by E17, MATRIX_0: 38;

            then ((Kx . i) . m) in ( rng (Kx . i)) by FUNCT_1: 3;

            then x in ( union ( rng (Kx . i))) by E18, TARSKI:def 4;

            hence x in ( Union (Kx . i)) by CARD_3:def 4;

          end;

          then ((F1 . i) /\ ( Union F2)) c= ( Union (Kx . i)) by TARSKI:def 3;

          then ((F1 . i) /\ ( Union F2)) = ( Union (Kx . i)) by E13, XBOOLE_0:def 10;

          then

           E19: (P . ((F1 . i) /\ ( Union F2))) = ( Sum (P * (Kx . i))) by E3, E4, E8, A3;

          

           E20: i in ( Seg ( len Mx)) by C1, E0, E1;

          

           E21: (Mx . i) = ( Line (Mx,i)) & (Kx . i) = ( Line (Kx,i)) by E5, MATRIX_0: 60;

          ( rng (Kx . i)) c= S;

          then ( rng (Kx . i)) c= ( dom P) by FUNCT_2:def 1;

          then

           E22: ( dom (P * (Kx . i))) = ( dom (Kx . i)) by RELAT_1: 27;

          then ( len (P * (Kx . i))) = ( len (Kx . i)) by FINSEQ_3: 29;

          then

           E23a: ( len (P * (Kx . i))) = ( width Kx) by E21, MATRIX_0:def 7;

          then

           E23: ( len (P * (Kx . i))) = ( len (Mx . i)) by C4, E21, MATRIX_0:def 7;

          for k be Nat st 1 <= k & k <= ( len (P * (Kx . i))) holds ((P * (Kx . i)) . k) = ((Mx . i) . k)

          proof

            let k be Nat;

            assume

             E24: 1 <= k & k <= ( len (P * (Kx . i)));

            then k in ( dom (Kx . i)) & k in ( dom (Mx . i)) by E23, E22, FINSEQ_3: 25;

            then

             E25: [i, k] in ( Indices Kx) & [i, k] in ( Indices Mx) by E5, MATRPROB: 13;

            k in ( dom (P * (Kx . i))) by E24, FINSEQ_3: 25;

            then ((P * (Kx . i)) . k) = (P . ((Kx . i) . k)) by FUNCT_1: 12;

            then ((P * (Kx . i)) . k) = (P . (Kx * (i,k))) by E25, MATRPROB: 14;

            then ((P * (Kx . i)) . k) = (P . ((F1 . i) /\ (F2 . k))) by E25, KX1;

            then ((P * (Kx . i)) . k) = (Mx * (i,k)) by E25, A2;

            hence ((P * (Kx . i)) . k) = ((Mx . i) . k) by E25, MATRPROB: 14;

          end;

          then

           E27: (P * (Kx . i)) = (Mx . i) by E23a, C4, E21, MATRIX_0:def 7;

          (F1 . i) c= ( union ( rng F1)) by E2, FUNCT_1: 3, ZFMISC_1: 74;

          then (F1 . i) c= ( Union F1) by CARD_3:def 4;

          then ((F1 . i) /\ ( Union F2)) = (F1 . i) by A1, XBOOLE_1: 28;

          then ((P * F1) . i) = ( Sum (P * (Kx . i))) by E2, E19, FUNCT_1: 13;

          hence ((P * F1) . i) = (( Sum Mx) . i) by E20, E27, E21, Th16;

        end;

      end;

      consider SMF1 be Function of NAT , ExtREAL such that

       A2: ( Sum (P * F1)) = (SMF1 . ( len (P * F1))) & (SMF1 . 0 ) = 0 & for i be Nat st i < ( len (P * F1)) holds (SMF1 . (i + 1)) = ((SMF1 . i) + ((P * F1) . (i + 1))) by EXTREAL1:def 2;

      consider LL be Function of NAT , ExtREAL such that

       C7: ( SumAll Mx) = (LL . ( len ( Sum Mx))) & (LL . 0 ) = 0. & for i be Nat st i < ( len ( Sum Mx)) holds (LL . (i + 1)) = ((LL . i) + (( Sum Mx) . (i + 1))) by EXTREAL1:def 2;

      defpred PK1[ Nat] means $1 <= ( len (P * F1)) implies (SMF1 . $1) = (LL . $1);

      

       C8: PK1[ 0 ] by A2, C7;

      

       C9: for i be Nat st PK1[i] holds PK1[(i + 1)]

      proof

        let i be Nat;

        assume

         V1: PK1[i];

        assume

         V3: (i + 1) <= ( len (P * F1));

        then (SMF1 . (i + 1)) = ((SMF1 . i) + ((P * F1) . (i + 1))) by A2, NAT_1: 13;

        then (SMF1 . (i + 1)) = ((LL . i) + (( Sum Mx) . (i + 1))) by C6, V1, V3, NAT_1: 13;

        hence (SMF1 . (i + 1)) = (LL . (i + 1)) by C7, V3, C2, NAT_1: 13;

      end;

      for i be Nat holds PK1[i] from NAT_1:sch 2( C8, C9);

      hence ( Sum (P * F1)) = ( SumAll Mx) by A2, C2, C7;

    end;

    theorem :: MEASURE9:43

    

     Th41: for X be set, S be with_empty_element cap-closed Subset-Family of X, F1,F2 be non empty disjoint_valued FinSequence of S, P be nonnegative zeroed Function of S, ExtREAL , Mx be Matrix of ( len F1), ( len F2), ExtREAL st ( Union F1) = ( Union F2) & (for i,j be Nat st [i, j] in ( Indices Mx) holds (Mx * (i,j)) = (P . ((F1 . i) /\ (F2 . j)))) & (for F be disjoint_valued FinSequence of S st ( Union F) in S holds (P . ( Union F)) = ( Sum (P * F))) holds (for i be Nat st i <= ( len (P * F2)) holds ((P * F2) . i) = (( Sum (Mx @ )) . i)) & ( Sum (P * F2)) = ( SumAll (Mx @ ))

    proof

      let X be set, S be with_empty_element cap-closed Subset-Family of X, F1,F2 be non empty disjoint_valued FinSequence of S, P be nonnegative zeroed Function of S, ExtREAL , Mx be Matrix of ( len F1), ( len F2), ExtREAL ;

      assume that

       A1: ( Union F1) = ( Union F2) and

       A2: for i,j be Nat st [i, j] in ( Indices Mx) holds (Mx * (i,j)) = (P . ((F1 . i) /\ (F2 . j))) and

       A3: for F be disjoint_valued FinSequence of S st ( Union F) in S holds (P . ( Union F)) = ( Sum (P * F));

      consider Kx be Matrix of ( len F1), ( len F2), S such that

       KX1: for i,j be Nat st [i, j] in ( Indices Kx) holds (Kx * (i,j)) = ((F1 . i) /\ (F2 . j)) by FStoMAT1;

      

       A5: ( len (P * F2)) = ( len F2) by FINSEQ_2: 33;

      

       C3: ( len Kx) = ( len F1) & ( len Mx) = ( len F1) by MATRIX_0:def 2;

      then ( width Kx) = ( len F2) & ( width Mx) = ( len F2) by MATRIX_0: 20;

      then

       C5: ( len (Kx @ )) = ( len F2) & ( len (Mx @ )) = ( len F2) & ( width (Kx @ )) = ( len F1) & ( width (Mx @ )) = ( len F1) by C3, MATRIX_0: 29;

      then

       D2: ( len (P * F2)) = ( len ( Sum (Mx @ ))) by A5, Def5;

      thus

       D6: for i be Nat st i <= ( len (P * F2)) holds ((P * F2) . i) = (( Sum (Mx @ )) . i)

      proof

        let i be Nat;

        assume

         E0: i <= ( len (P * F2));

        per cases ;

          suppose i = 0 ;

          then not i in ( dom (P * F2)) & not i in ( dom ( Sum (Mx @ ))) by FINSEQ_3: 24;

          then ((P * F2) . i) = 0 & (( Sum (Mx @ )) . i) = 0 by FUNCT_1:def 2;

          hence ((P * F2) . i) = (( Sum (Mx @ )) . i);

        end;

          suppose i <> 0 ;

          then

           E1: 1 <= i by NAT_1: 14;

          then i in ( dom (P * F2)) by E0, FINSEQ_3: 25;

          then

           E2: i in ( dom F2) by FUNCT_1: 11;

          then (F2 . i) c= ( union ( rng F2)) by FUNCT_1: 3, ZFMISC_1: 74;

          then (F2 . i) c= ( Union F1) by A1, CARD_3:def 4;

          then

           E3: ((F2 . i) /\ ( Union F1)) = (F2 . i) by XBOOLE_1: 28;

          

           E4: (F2 . i) in ( rng F2) by E2, FUNCT_1: 3;

          

           E5: i in ( dom (Kx @ )) & i in ( dom (Mx @ )) by C5, A5, E0, E1, FINSEQ_3: 25;

          for p,q be object st p <> q holds (((Kx @ ) . i) . p) misses (((Kx @ ) . i) . q)

          proof

            let p,q be object;

            assume

             SA0: p <> q;

            per cases ;

              suppose

               SA1: p in ( dom ((Kx @ ) . i)) & q in ( dom ((Kx @ ) . i));

              then

              reconsider p1 = p, q1 = q as Nat;

              

               E6: [i, p1] in ( Indices (Kx @ )) & [i, q1] in ( Indices (Kx @ )) by SA1, E5, MATRIX_0: 37;

              then

               EE6: [p1, i] in ( Indices Kx) & [q1, i] in ( Indices Kx) by MATRIX_0:def 6;

              ((Kx @ ) * (i,p1)) = (((Kx @ ) . i) . p) & ((Kx @ ) * (i,q1)) = (((Kx @ ) . i) . q) by E6, MATRPROB: 14;

              then (((Kx @ ) . i) . p) = (Kx * (p1,i)) & (((Kx @ ) . i) . q) = (Kx * (q1,i)) by EE6, MATRIX_0:def 6;

              then (((Kx @ ) . i) . p) = ((F2 . i) /\ (F1 . p1)) & (((Kx @ ) . i) . q) = ((F2 . i) /\ (F1 . q1)) by EE6, KX1;

              hence (((Kx @ ) . i) . p) misses (((Kx @ ) . i) . q) by SA0, PROB_2:def 2, XBOOLE_1: 76;

            end;

              suppose not p in ( dom ((Kx @ ) . i));

              then (((Kx @ ) . i) . p) = {} by FUNCT_1:def 2;

              hence (((Kx @ ) . i) . p) misses (((Kx @ ) . i) . q) by XBOOLE_1: 65;

            end;

              suppose not q in ( dom ((Kx @ ) . i));

              then (((Kx @ ) . i) . q) = {} by FUNCT_1:def 2;

              hence (((Kx @ ) . i) . p) misses (((Kx @ ) . i) . q) by XBOOLE_1: 65;

            end;

          end;

          then

           E8: ((Kx @ ) . i) is disjoint_valued FinSequence of S by PROB_2:def 2;

          now

            let x be object;

            assume x in ( Union ((Kx @ ) . i));

            then x in ( union ( rng ((Kx @ ) . i))) by CARD_3:def 4;

            then

            consider A be set such that

             E9: x in A & A in ( rng ((Kx @ ) . i)) by TARSKI:def 4;

            consider m be object such that

             E10: m in ( dom ((Kx @ ) . i)) & A = (((Kx @ ) . i) . m) by E9, FUNCT_1:def 3;

            reconsider m as Nat by E10;

            

             E11: [i, m] in ( Indices (Kx @ )) by E10, E5, MATRIX_0: 37;

            then

             EE11: [m, i] in ( Indices Kx) by MATRIX_0:def 6;

            (((Kx @ ) . i) . m) = ((Kx @ ) * (i,m)) by E11, MATRPROB: 14;

            then (((Kx @ ) . i) . m) = (Kx * (m,i)) by EE11, MATRIX_0:def 6;

            then (((Kx @ ) . i) . m) = ((F2 . i) /\ (F1 . m)) by EE11, KX1;

            then

             E12: x in (F2 . i) & x in (F1 . m) by E9, E10, XBOOLE_0:def 4;

            1 <= m & m <= ( len F1) by EE11, MATRIX_0: 33;

            then m in ( dom F1) by FINSEQ_3: 25;

            then (F1 . m) in ( rng F1) by FUNCT_1: 3;

            then x in ( union ( rng F1)) by E12, TARSKI:def 4;

            then x in ( Union F1) by CARD_3:def 4;

            hence x in ((F2 . i) /\ ( Union F1)) by E12, XBOOLE_0:def 4;

          end;

          then

           E13: ( Union ((Kx @ ) . i)) c= ((F2 . i) /\ ( Union F1)) by TARSKI:def 3;

          now

            let x be object;

            assume x in ((F2 . i) /\ ( Union F1));

            then

             E14: x in (F2 . i) & x in ( Union F1) by XBOOLE_0:def 4;

            then x in ( union ( rng F1)) by CARD_3:def 4;

            then

            consider A be set such that

             E15: x in A & A in ( rng F1) by TARSKI:def 4;

            consider m be object such that

             E16: m in ( dom F1) & A = (F1 . m) by E15, FUNCT_1:def 3;

            reconsider m as Nat by E16;

            1 <= i & i <= ( len F2) & 1 <= m & m <= ( len F1) by E2, E16, FINSEQ_3: 25;

            then

             EE17: [m, i] in ( Indices Kx) by MATRIX_0: 31;

            then

             E17: [i, m] in ( Indices (Kx @ )) by MATRIX_0:def 6;

            (((Kx @ ) . i) . m) = ((Kx @ ) * (i,m)) by E17, MATRPROB: 14;

            then (((Kx @ ) . i) . m) = (Kx * (m,i)) by EE17, MATRIX_0:def 6;

            then (((Kx @ ) . i) . m) = ((F2 . i) /\ (F1 . m)) by EE17, KX1;

            then

             E18: x in (((Kx @ ) . i) . m) by E14, E15, E16, XBOOLE_0:def 4;

            m in ( dom ((Kx @ ) . i)) by E17, MATRIX_0: 38;

            then (((Kx @ ) . i) . m) in ( rng ((Kx @ ) . i)) by FUNCT_1: 3;

            then x in ( union ( rng ((Kx @ ) . i))) by E18, TARSKI:def 4;

            hence x in ( Union ((Kx @ ) . i)) by CARD_3:def 4;

          end;

          then ((F2 . i) /\ ( Union F1)) c= ( Union ((Kx @ ) . i)) by TARSKI:def 3;

          then ((F2 . i) /\ ( Union F1)) = ( Union ((Kx @ ) . i)) by E13, XBOOLE_0:def 10;

          then

           E19: (P . ((F2 . i) /\ ( Union F1))) = ( Sum (P * ((Kx @ ) . i))) by E3, E4, E8, A3;

          

           E20: i in ( Seg ( len (Mx @ ))) by C5, A5, E0, E1;

          

           E21: ((Mx @ ) . i) = ( Line ((Mx @ ),i)) & ((Kx @ ) . i) = ( Line ((Kx @ ),i)) by E5, MATRIX_0: 60;

          ( rng ((Kx @ ) . i)) c= S;

          then ( rng ((Kx @ ) . i)) c= ( dom P) by FUNCT_2:def 1;

          then

           E22: ( dom (P * ((Kx @ ) . i))) = ( dom ((Kx @ ) . i)) by RELAT_1: 27;

          then ( len (P * ((Kx @ ) . i))) = ( len ((Kx @ ) . i)) by FINSEQ_3: 29;

          then

           F23: ( len (P * ((Kx @ ) . i))) = ( width (Kx @ )) by E21, MATRIX_0:def 7;

          then

           E23: ( len (P * ((Kx @ ) . i))) = ( len ((Mx @ ) . i)) by C5, E21, MATRIX_0:def 7;

          for k be Nat st 1 <= k & k <= ( len (P * ((Kx @ ) . i))) holds ((P * ((Kx @ ) . i)) . k) = (((Mx @ ) . i) . k)

          proof

            let k be Nat;

            assume

             E24: 1 <= k & k <= ( len (P * ((Kx @ ) . i)));

            then k in ( dom ((Kx @ ) . i)) & k in ( dom ((Mx @ ) . i)) by E23, E22, FINSEQ_3: 25;

            then

             E25: [i, k] in ( Indices (Kx @ )) & [i, k] in ( Indices (Mx @ )) by E5, MATRPROB: 13;

            then

             EE25: [k, i] in ( Indices Kx) & [k, i] in ( Indices Mx) by MATRIX_0:def 6;

            k in ( dom (P * ((Kx @ ) . i))) by E24, FINSEQ_3: 25;

            then ((P * ((Kx @ ) . i)) . k) = (P . (((Kx @ ) . i) . k)) by FUNCT_1: 12;

            then ((P * ((Kx @ ) . i)) . k) = (P . ((Kx @ ) * (i,k))) by E25, MATRPROB: 14;

            then ((P * ((Kx @ ) . i)) . k) = (P . (Kx * (k,i))) by EE25, MATRIX_0:def 6;

            then ((P * ((Kx @ ) . i)) . k) = (P . ((F2 . i) /\ (F1 . k))) by EE25, KX1;

            then ((P * ((Kx @ ) . i)) . k) = (Mx * (k,i)) by EE25, A2;

            then ((P * ((Kx @ ) . i)) . k) = ((Mx @ ) * (i,k)) by EE25, MATRIX_0:def 6;

            hence ((P * ((Kx @ ) . i)) . k) = (((Mx @ ) . i) . k) by E25, MATRPROB: 14;

          end;

          then

           E27: (P * ((Kx @ ) . i)) = ((Mx @ ) . i) by F23, C5, E21, MATRIX_0:def 7;

          (F2 . i) c= ( union ( rng F2)) by E2, FUNCT_1: 3, ZFMISC_1: 74;

          then (F2 . i) c= ( Union F2) by CARD_3:def 4;

          then ((F2 . i) /\ ( Union F1)) = (F2 . i) by A1, XBOOLE_1: 28;

          then ((P * F2) . i) = ( Sum (P * ((Kx @ ) . i))) by E2, E19, FUNCT_1: 13;

          hence ((P * F2) . i) = (( Sum (Mx @ )) . i) by E20, E27, E21, Th16;

        end;

      end;

      consider SMF2 be Function of NAT , ExtREAL such that

       A3: ( Sum (P * F2)) = (SMF2 . ( len (P * F2))) & (SMF2 . 0 ) = 0 & for i be Nat st i < ( len (P * F2)) holds (SMF2 . (i + 1)) = ((SMF2 . i) + ((P * F2) . (i + 1))) by EXTREAL1:def 2;

      consider LL be Function of NAT , ExtREAL such that

       D7: ( SumAll (Mx @ )) = (LL . ( len ( Sum (Mx @ )))) & (LL . 0 ) = 0. & for i be Nat st i < ( len ( Sum (Mx @ ))) holds (LL . (i + 1)) = ((LL . i) + (( Sum (Mx @ )) . (i + 1))) by EXTREAL1:def 2;

      defpred PK2[ Nat] means $1 <= ( len (P * F2)) implies (SMF2 . $1) = (LL . $1);

      

       D8: PK2[ 0 ] by A3, D7;

      

       D9: for i be Nat st PK2[i] holds PK2[(i + 1)]

      proof

        let i be Nat;

        assume

         V1: PK2[i];

        assume

         V3: (i + 1) <= ( len (P * F2));

        then (SMF2 . (i + 1)) = ((SMF2 . i) + ((P * F2) . (i + 1))) by A3, NAT_1: 13;

        then (SMF2 . (i + 1)) = ((LL . i) + (( Sum (Mx @ )) . (i + 1))) by D6, V1, V3, NAT_1: 13;

        hence (SMF2 . (i + 1)) = (LL . (i + 1)) by D7, V3, D2, NAT_1: 13;

      end;

      for i be Nat holds PK2[i] from NAT_1:sch 2( D8, D9);

      hence ( Sum (P * F2)) = ( SumAll (Mx @ )) by A3, D2, D7;

    end;

    theorem :: MEASURE9:44

    

     Th42: for X be set, S be with_empty_element semi-diff-closed cap-closed Subset-Family of X, P be pre-Measure of S, A be set st A in ( Ring_generated_by S) holds for F1,F2 be disjoint_valued FinSequence of S st A = ( Union F1) & A = ( Union F2) holds ( Sum (P * F1)) = ( Sum (P * F2))

    proof

      let X be set, S be with_empty_element semi-diff-closed cap-closed Subset-Family of X, P be pre-Measure of S, A be set;

      assume A in ( Ring_generated_by S);

      hereby

        let F1,F2 be disjoint_valued FinSequence of S;

        assume

         A1: A = ( Union F1) & A = ( Union F2);

        consider SMF1 be Function of NAT , ExtREAL such that

         A2: ( Sum (P * F1)) = (SMF1 . ( len (P * F1))) & (SMF1 . 0 ) = 0 & for i be Nat st i < ( len (P * F1)) holds (SMF1 . (i + 1)) = ((SMF1 . i) + ((P * F1) . (i + 1))) by EXTREAL1:def 2;

        consider SMF2 be Function of NAT , ExtREAL such that

         A3: ( Sum (P * F2)) = (SMF2 . ( len (P * F2))) & (SMF2 . 0 ) = 0 & for i be Nat st i < ( len (P * F2)) holds (SMF2 . (i + 1)) = ((SMF2 . i) + ((P * F2) . (i + 1))) by EXTREAL1:def 2;

        ( dom P) = S by FUNCT_2:def 1;

        then ( rng F1) c= ( dom P) & ( rng F2) c= ( dom P);

        then

         A4: ( dom (P * F1)) = ( dom F1) & ( dom (P * F2)) = ( dom F2) by RELAT_1: 27;

        then

         A5: ( dom (P * F1)) = ( Seg ( len F1)) & ( dom (P * F2)) = ( Seg ( len F2)) & ( len (P * F1)) = ( len F1) & ( len (P * F2)) = ( len F2) by FINSEQ_1:def 3, FINSEQ_3: 29;

        per cases ;

          suppose

           A6: ( len (P * F1)) = 0 ;

          then (P * F1) = {} ;

          then F1 = {} by A4;

          then ( rng F1) = {} ;

          then ( Union F2) = {} by A1, CARD_3:def 4, ZFMISC_1: 2;

          then

           G7: ( union ( rng F2)) = {} by CARD_3:def 4;

          defpred S[ Nat] means $1 <= ( len (P * F2)) implies (SMF2 . $1) = 0 ;

          

           A8: S[ 0 ] by A3;

          

           A9: for i be Nat st S[i] holds S[(i + 1)]

          proof

            let i be Nat;

            assume

             A10: S[i];

            assume

             A11: (i + 1) <= ( len (P * F2));

            then

             A13: (SMF2 . (i + 1)) = ((SMF2 . i) + ((P * F2) . (i + 1))) & (SMF2 . i) = 0 by A3, A10, NAT_1: 13;

            

             A14: (i + 1) in ( dom (P * F2)) by A11, NAT_1: 11, FINSEQ_3: 25;

            then (F2 . (i + 1)) = {} by A4, G7, ORDERS_1: 6, FUNCT_1: 3;

            then (P . (F2 . (i + 1))) = 0 by VALUED_0:def 19;

            then ((P * F2) . (i + 1)) = 0 by A14, FUNCT_1: 12;

            hence (SMF2 . (i + 1)) = 0 by A13;

          end;

          for i be Nat holds S[i] from NAT_1:sch 2( A8, A9);

          hence ( Sum (P * F1)) = ( Sum (P * F2)) by A2, A3, A6;

        end;

          suppose

           B6: ( len (P * F2)) = 0 ;

          then (P * F2) = {} ;

          then F2 = {} by A4;

          then ( rng F2) = {} ;

          then ( Union F1) = {} by A1, CARD_3:def 4, ZFMISC_1: 2;

          then

           E7: ( union ( rng F1)) = {} by CARD_3:def 4;

          defpred S[ Nat] means $1 <= ( len (P * F1)) implies (SMF1 . $1) = 0 ;

          

           B8: S[ 0 ] by A2;

          

           B9: for i be Nat st S[i] holds S[(i + 1)]

          proof

            let i be Nat;

            assume

             B10: S[i];

            assume

             B11: (i + 1) <= ( len (P * F1));

            then

             B13: (SMF1 . (i + 1)) = ((SMF1 . i) + ((P * F1) . (i + 1))) & (SMF1 . i) = 0 by A2, B10, NAT_1: 13;

            

             B14: (i + 1) in ( dom (P * F1)) by B11, NAT_1: 11, FINSEQ_3: 25;

            then (F1 . (i + 1)) = {} by A4, E7, ORDERS_1: 6, FUNCT_1: 3;

            then (P . (F1 . (i + 1))) = 0 by VALUED_0:def 19;

            then ((P * F1) . (i + 1)) = 0 by B14, FUNCT_1: 12;

            hence (SMF1 . (i + 1)) = 0 by B13;

          end;

          for i be Nat holds S[i] from NAT_1:sch 2( B8, B9);

          hence ( Sum (P * F1)) = ( Sum (P * F2)) by A2, A3, B6;

        end;

          suppose

           A15: ( len (P * F1)) <> 0 & ( len (P * F2)) <> 0 ;

          defpred Mx[ Nat, Nat, set] means $3 = (P . ((F1 . $1) /\ (F2 . $2)));

          

           MX0: for i,j be Nat st [i, j] in [:( Seg ( len F1)), ( Seg ( len F2)):] holds ex A be Element of ExtREAL st Mx[i, j, A];

          consider Mx be Matrix of ( len F1), ( len F2), ExtREAL such that

           MX1: for i,j be Nat st [i, j] in ( Indices Mx) holds Mx[i, j, (Mx * (i,j))] from MATRIX_0:sch 2( MX0);

          

           C3: ( len Mx) = ( len F1) by MATRIX_0:def 2;

          then

           C4: ( width Mx) = ( len F2) by A15, A5, MATRIX_0: 20;

          

           CC0: for F be disjoint_valued FinSequence of S st ( Union F) in S holds (P . ( Union F)) = ( Sum (P * F)) by Def8;

          

           C0: F1 is non empty & F2 is non empty by A15;

          then

           C10: ( Sum (P * F1)) = ( SumAll Mx) by A1, MX1, CC0, Th40;

          

           D10: ( Sum (P * F2)) = ( SumAll (Mx @ )) by C0, A1, MX1, CC0, Th41;

          for i be Nat st i in ( dom Mx) holds not -infty in ( rng (Mx . i))

          proof

            let i be Nat;

            assume

             F1: i in ( dom Mx);

            assume -infty in ( rng (Mx . i));

            then

            consider j be object such that

             F2: j in ( dom (Mx . i)) & ((Mx . i) . j) = -infty by FUNCT_1:def 3;

            reconsider j as Nat by F2;

            

             F3: [i, j] in ( Indices Mx) by F1, F2, MATRPROB: 13;

            then ((Mx . i) . j) = (Mx * (i,j)) by MATRPROB: 14;

            then

             F5: ((Mx . i) . j) = (P . ((F1 . i) /\ (F2 . j))) by F3, MX1;

            i in ( Seg ( len Mx)) & j in ( Seg ( width Mx)) by F3, MATRPROB: 12;

            then i in ( dom F1) & j in ( dom F2) by C3, C4, FINSEQ_1:def 3;

            then (F1 . i) in ( rng F1) & (F2 . j) in ( rng F2) by FUNCT_1: 3;

            then ((F1 . i) /\ (F2 . j)) in S by FINSUB_1:def 2;

            hence contradiction by F2, F5, MEASURE1:def 2;

          end;

          hence ( Sum (P * F1)) = ( Sum (P * F2)) by C10, D10, Th28;

        end;

      end;

    end;

    theorem :: MEASURE9:45

    

     Th43: for f1,f2 be FinSequence st f1 is disjoint_valued & f2 is disjoint_valued & ( union ( rng f1)) misses ( union ( rng f2)) holds (f1 ^ f2) is disjoint_valued

    proof

      let f1,f2 be FinSequence;

      assume that

       A1: f1 is disjoint_valued & f2 is disjoint_valued and

       A2: ( union ( rng f1)) misses ( union ( rng f2));

      now

        let x,y be object;

        assume

         A3: x <> y;

        per cases ;

          suppose

           A4: x in ( dom (f1 ^ f2)) & y in ( dom (f1 ^ f2));

          then

          reconsider x1 = x, y1 = y as Nat;

          per cases by A4, FINSEQ_1: 25;

            suppose x1 in ( dom f1) & y1 in ( dom f1);

            then ((f1 ^ f2) . x) = (f1 . x) & ((f1 ^ f2) . y) = (f1 . y) by FINSEQ_1:def 7;

            hence ((f1 ^ f2) . x) misses ((f1 ^ f2) . y) by A1, A3, PROB_2:def 2;

          end;

            suppose

             A6: x1 in ( dom f1) & ex n be Nat st n in ( dom f2) & y1 = (( len f1) + n);

            then

            consider n be Nat such that

             A7: n in ( dom f2) & y1 = (( len f1) + n);

            ((f1 ^ f2) . x) = (f1 . x) by A6, FINSEQ_1:def 7;

            then

             A8: ((f1 ^ f2) . x) in ( rng f1) by A6, FUNCT_1: 3;

            ((f1 ^ f2) . y) = (f2 . n) by A7, FINSEQ_1:def 7;

            then

             A9: ((f1 ^ f2) . y) in ( rng f2) by A7, FUNCT_1: 3;

            now

              assume ((f1 ^ f2) . x) meets ((f1 ^ f2) . y);

              then

              consider z be object such that

               A10: z in ((f1 ^ f2) . x) & z in ((f1 ^ f2) . y) by XBOOLE_0: 3;

              z in ( union ( rng f1)) & z in ( union ( rng f2)) by A8, A9, A10, TARSKI:def 4;

              hence contradiction by A2, XBOOLE_0: 3;

            end;

            hence ((f1 ^ f2) . x) misses ((f1 ^ f2) . y);

          end;

            suppose

             A11: y1 in ( dom f1) & ex n be Nat st n in ( dom f2) & x1 = (( len f1) + n);

            then

            consider n be Nat such that

             A12: n in ( dom f2) & x1 = (( len f1) + n);

            ((f1 ^ f2) . x) = (f2 . n) by A12, FINSEQ_1:def 7;

            then

             A13: ((f1 ^ f2) . x) in ( rng f2) by A12, FUNCT_1: 3;

            ((f1 ^ f2) . y) = (f1 . y) by A11, FINSEQ_1:def 7;

            then

             A14: ((f1 ^ f2) . y) in ( rng f1) by A11, FUNCT_1: 3;

            now

              assume ((f1 ^ f2) . x) meets ((f1 ^ f2) . y);

              then

              consider z be object such that

               A15: z in ((f1 ^ f2) . x) & z in ((f1 ^ f2) . y) by XBOOLE_0: 3;

              z in ( union ( rng f1)) & z in ( union ( rng f2)) by A13, A14, A15, TARSKI:def 4;

              hence contradiction by A2, XBOOLE_0: 3;

            end;

            hence ((f1 ^ f2) . x) misses ((f1 ^ f2) . y);

          end;

            suppose

             A16: (ex n be Nat st n in ( dom f2) & x1 = (( len f1) + n)) & (ex m be Nat st m in ( dom f2) & y1 = (( len f1) + m));

            then

            consider n be Nat such that

             A17: n in ( dom f2) & x1 = (( len f1) + n);

            

             A18: ((f1 ^ f2) . x) = (f2 . n) by A17, FINSEQ_1:def 7;

            consider m be Nat such that

             A19: m in ( dom f2) & y1 = (( len f1) + m) by A16;

            ((f1 ^ f2) . y) = (f2 . m) by A19, FINSEQ_1:def 7;

            hence ((f1 ^ f2) . x) misses ((f1 ^ f2) . y) by A1, A18, A17, A19, A3, PROB_2:def 2;

          end;

        end;

          suppose not x in ( dom (f1 ^ f2)) or not y in ( dom (f1 ^ f2));

          then ((f1 ^ f2) . x) = {} or ((f1 ^ f2) . y) = {} by FUNCT_1:def 2;

          hence ((f1 ^ f2) . x) misses ((f1 ^ f2) . y) by XBOOLE_1: 65;

        end;

      end;

      hence (f1 ^ f2) is disjoint_valued by PROB_2:def 2;

    end;

    theorem :: MEASURE9:46

    for X be set, P be with_empty_element semi-diff-closed Subset-Family of X, M be pre-Measure of P, A,B be set st A in P & B in P & (A \ B) in P & B c= A holds (M . A) >= (M . B)

    proof

      let X be set, P be with_empty_element semi-diff-closed Subset-Family of X, M be pre-Measure of P, A,B be set;

      assume that

       A1: A in P & B in P & (A \ B) in P and

       A2: B c= A;

      consider F be disjoint_valued FinSequence of P such that

       A3: (A \ B) = ( Union F) by A1, SRINGS_3:def 1;

      

       A7: ( rng <*B*>) = {B} by FINSEQ_1: 38;

      then

      reconsider G = <*B*> as disjoint_valued FinSequence of P by FINSEQ_1:def 4, A1, ZFMISC_1: 31;

      now

        assume ( union ( rng G)) meets ( union ( rng F));

        then

        consider x be object such that

         A4: x in ( union ( rng G)) & x in ( union ( rng F)) by XBOOLE_0: 3;

        consider P be set such that

         A5: x in P & P in ( rng G) by A4, TARSKI:def 4;

        P in {B} by A5, FINSEQ_1: 38;

        then

         A6: x in B by A5, TARSKI:def 1;

        x in (A \ B) by A3, A4, CARD_3:def 4;

        hence contradiction by A6, XBOOLE_0:def 5;

      end;

      then

      reconsider H = (G ^ F) as disjoint_valued FinSequence of P by Th43;

      

       A8: ( union ( rng G)) = B by A7, ZFMISC_1: 25;

      ( rng H) = (( rng G) \/ ( rng F)) by FINSEQ_1: 31;

      then ( union ( rng H)) = (( union ( rng G)) \/ ( union ( rng F))) by ZFMISC_1: 78;

      

      then ( Union H) = (B \/ ( union ( rng F))) by A8, CARD_3:def 4

      .= (B \/ (A \ B)) by A3, CARD_3:def 4;

      then ( Union H) = (A \/ B) by XBOOLE_1: 39;

      then ( Union H) = A by A2, XBOOLE_1: 12;

      then

       A9: (M . A) = ( Sum (M * H)) by A1, Def8;

      ( Union G) = B by A8, CARD_3:def 4;

      then

       A10: (M . B) = ( Sum (M * G)) by A1, Def8;

       B0:

      now

        assume -infty in ( rng (M * G));

        then

        consider n be Element of NAT such that

         B1: n in ( dom (M * G)) & -infty = ((M * G) . n) by PARTFUN1: 3;

        (M . (G . n)) = -infty by B1, FUNCT_1: 12;

        hence contradiction by SUPINF_2: 51;

      end;

       A11:

      now

        assume -infty in ( rng (M * F));

        then

        consider n be Element of NAT such that

         B2: n in ( dom (M * F)) & -infty = ((M * F) . n) by PARTFUN1: 3;

        (M . (F . n)) = -infty by B2, FUNCT_1: 12;

        hence contradiction by SUPINF_2: 51;

      end;

       A12:

      now

        let n be Nat;

        assume n in ( dom (M * F));

        then ((M * F) . n) = (M . (F . n)) & (F . n) in ( dom M) by FUNCT_1: 11, FUNCT_1: 12;

        hence ((M * F) . n) >= 0 by SUPINF_2: 51;

      end;

      (M * H) = ((M * G) ^ (M * F)) by FINSEQOP: 9;

      then ( Sum (M * H)) = (( Sum (M * G)) + ( Sum (M * F))) by A11, B0, EXTREAL1: 10;

      hence (M . B) <= (M . A) by A9, A10, A12, MESFUNC5: 53, XXREAL_3: 39;

    end;

    theorem :: MEASURE9:47

    

     Th45: for Y,S be non empty set, F be PartFunc of Y, S, M be Function of S, ExtREAL st M is nonnegative holds (M * F) is nonnegative

    proof

      let Y,S be non empty set;

      let F be PartFunc of Y, S;

      let M be Function of S, ExtREAL ;

      assume

       A1: M is nonnegative;

      now

        let n be object;

        per cases ;

          suppose n in ( dom (M * F));

          then ((M * F) . n) = (M . (F . n)) by FUNCT_1: 12;

          hence ((M * F) . n) >= 0 by A1, SUPINF_2: 51;

        end;

          suppose not n in ( dom (M * F));

          hence ((M * F) . n) >= 0 by FUNCT_1:def 2;

        end;

      end;

      hence thesis by SUPINF_2: 51;

    end;

    theorem :: MEASURE9:48

    

     Th46: for X be set, S be with_empty_element semi-diff-closed cap-closed Subset-Family of X, P be pre-Measure of S holds ex M be nonnegative additive zeroed Function of ( Ring_generated_by S), ExtREAL st for A be set st A in ( Ring_generated_by S) holds for F be disjoint_valued FinSequence of S st A = ( Union F) holds (M . A) = ( Sum (P * F))

    proof

      let X be set, S be with_empty_element semi-diff-closed cap-closed Subset-Family of X, P be pre-Measure of S;

      defpred P[ object, object] means for F be disjoint_valued FinSequence of S st $1 = ( Union F) holds $2 = ( Sum (P * F));

      

       A1: for A be object st A in ( Ring_generated_by S) holds ex p be object st p in ExtREAL & P[A, p]

      proof

        let A be object;

        assume

         A2: A in ( Ring_generated_by S);

        then A in ( DisUnion S) by SRINGS_3: 18;

        then

        consider V be Subset of X such that

         A3: A = V & ex F be disjoint_valued FinSequence of S st V = ( Union F);

        consider F be disjoint_valued FinSequence of S such that

         A4: V = ( Union F) by A3;

        set p = ( Sum (P * F));

        take p;

        thus p in ExtREAL & P[A, p] by A2, A3, A4, Th42;

      end;

      consider M be Function of ( Ring_generated_by S), ExtREAL such that

       A5: for A be object st A in ( Ring_generated_by S) holds P[A, (M . A)] from FUNCT_2:sch 1( A1);

      

       A18: for A be Element of ( Ring_generated_by S) holds 0 <= (M . A)

      proof

        let A be Element of ( Ring_generated_by S);

        A in ( Ring_generated_by S);

        then A in ( DisUnion S) by SRINGS_3: 18;

        then

        consider V be Subset of X such that

         A7: A = V & ex F be disjoint_valued FinSequence of S st V = ( Union F);

        consider F be disjoint_valued FinSequence of S such that

         A8: V = ( Union F) by A7;

        consider PF be sequence of ExtREAL such that

         A10: ( Sum (P * F)) = (PF . ( len (P * F))) & (PF . 0 ) = 0. & for i be Nat st i < ( len (P * F)) holds (PF . (i + 1)) = ((PF . i) + ((P * F) . (i + 1))) by EXTREAL1:def 2;

        defpred P2[ Nat] means $1 <= ( len (P * F)) implies (PF . $1) >= 0 ;

        

         A11: P2[ 0 ] by A10;

        

         A12: for i be Nat st P2[i] holds P2[(i + 1)]

        proof

          let i be Nat;

          assume

           A13: P2[i];

          assume

           A14: (i + 1) <= ( len (P * F));

          then (i + 1) in ( dom (P * F)) by NAT_1: 11, FINSEQ_3: 25;

          then ((P * F) . (i + 1)) = (P . (F . (i + 1))) by FUNCT_1: 12;

          then

           A17: ((P * F) . (i + 1)) >= 0 by SUPINF_2: 51;

          (PF . (i + 1)) = ((PF . i) + ((P * F) . (i + 1))) by A14, A10, NAT_1: 13;

          hence (PF . (i + 1)) >= 0 by A13, A14, A17, NAT_1: 13;

        end;

        for i be Nat holds P2[i] from NAT_1:sch 2( A11, A12);

        then ( Sum (P * F)) >= 0 by A10;

        hence 0 <= (M . A) by A7, A8, A5;

      end;

      for A,B be Element of ( Ring_generated_by S) st A misses B & (A \/ B) in ( Ring_generated_by S) holds (M . (A \/ B)) = ((M . A) + (M . B))

      proof

        let A,B be Element of ( Ring_generated_by S);

        assume

         A19: A misses B & (A \/ B) in ( Ring_generated_by S);

        A in ( Ring_generated_by S);

        then A in ( DisUnion S) by SRINGS_3: 18;

        then

        consider V be Subset of X such that

         A20: A = V & ex F be disjoint_valued FinSequence of S st V = ( Union F);

        consider F be disjoint_valued FinSequence of S such that

         A21: V = ( Union F) by A20;

        B in ( Ring_generated_by S);

        then B in ( DisUnion S) by SRINGS_3: 18;

        then

        consider W be Subset of X such that

         A22: B = W & ex G be disjoint_valued FinSequence of S st W = ( Union G);

        consider G be disjoint_valued FinSequence of S such that

         A23: W = ( Union G) by A22;

        set H = (F ^ G);

        

         A24: A = ( union ( rng F)) & B = ( union ( rng G)) by A20, A21, A22, A23, CARD_3:def 4;

        then

        reconsider H as disjoint_valued FinSequence of S by A19, Th43;

        ( rng H) = (( rng F) \/ ( rng G)) by FINSEQ_1: 31;

        then ( union ( rng H)) = (( union ( rng F)) \/ ( union ( rng G))) by ZFMISC_1: 78;

        then (A \/ B) = ( Union H) by A24, CARD_3:def 4;

        then

         A25: (M . (A \/ B)) = ( Sum (P * H)) by A5;

        

         A26: (M . A) = ( Sum (P * F)) & (M . B) = ( Sum (P * G)) by A20, A21, A22, A23, A5;

        (P * F) is nonnegative by Th45;

        then

         A27: not -infty in ( rng (P * F)) by SUPINF_2:def 9, SUPINF_2:def 12;

        (P * G) is nonnegative by Th45;

        then

         A28: not -infty in ( rng (P * G)) by SUPINF_2:def 9, SUPINF_2:def 12;

        (P * H) = ((P * F) ^ (P * G)) by FINSEQOP: 9;

        hence (M . (A \/ B)) = ((M . A) + (M . B)) by A25, A26, A27, A28, EXTREAL1: 10;

      end;

      then

       A29: M is additive by MEASURE1:def 3;

      reconsider E = {} as Element of S by SETFAM_1:def 8;

      reconsider F = <*E*> as disjoint_valued FinSequence of S;

      ( rng F) = { {} } by FINSEQ_1: 38;

      then ( union ( rng F)) = {} by ZFMISC_1: 25;

      then ( Union F) = {} by CARD_3:def 4;

      then (M . {} ) = ( Sum (P * F)) by A5, FINSUB_1: 7;

      then (M . {} ) = ( Sum <*(P . {} )*>) by FINSEQ_2: 35;

      then (M . {} ) = (P . {} ) by EXTREAL1: 8;

      then (M . {} ) = 0 by VALUED_0:def 19;

      then

      reconsider M as nonnegative additive zeroed Function of ( Ring_generated_by S), ExtREAL by A18, A29, VALUED_0:def 19, MEASURE1:def 2;

      take M;

      thus thesis by A5;

    end;

    theorem :: MEASURE9:49

    for X,Y be set, F,G be Function of NAT , ( bool X) st (for i be Nat holds (G . i) = ((F . i) /\ Y)) & ( Union F) = Y holds ( Union G) = ( Union F)

    proof

      let X,Y be set, F,G be Function of NAT , ( bool X);

      assume that

       A1: for i be Nat holds (G . i) = ((F . i) /\ Y) and

       A2: ( Union F) = Y;

      now

        let x be object;

        assume x in ( Union G);

        then x in ( union ( rng G)) by CARD_3:def 4;

        then

        consider A be set such that

         A3: x in A & A in ( rng G) by TARSKI:def 4;

        consider i be Element of NAT such that

         A4: A = (G . i) by A3, FUNCT_2: 113;

        ( dom F) = NAT by FUNCT_2:def 1;

        then A = ((F . i) /\ Y) & i in ( dom F) by A1, A4;

        then x in (F . i) & (F . i) in ( rng F) by A3, XBOOLE_0:def 4, FUNCT_1: 3;

        then x in ( union ( rng F)) by TARSKI:def 4;

        hence x in ( Union F) by CARD_3:def 4;

      end;

      then

       A5: ( Union G) c= ( Union F) by TARSKI:def 3;

      now

        let x be object;

        assume

         A6: x in ( Union F);

        then x in ( union ( rng F)) by CARD_3:def 4;

        then

        consider A be set such that

         A7: x in A & A in ( rng F) by TARSKI:def 4;

        consider i be Element of NAT such that

         A8: A = (F . i) by A7, FUNCT_2: 113;

        ( dom G) = NAT by FUNCT_2:def 1;

        then x in ((F . i) /\ Y) & i in ( dom G) by A2, A6, A7, A8, XBOOLE_0:def 4;

        then x in (G . i) & (G . i) in ( rng G) by A1, FUNCT_1: 3;

        then x in ( union ( rng G)) by TARSKI:def 4;

        hence x in ( Union G) by CARD_3:def 4;

      end;

      then ( Union F) c= ( Union G) by TARSKI:def 3;

      hence thesis by A5, XBOOLE_0:def 10;

    end;

    theorem :: MEASURE9:50

    for X be set, S be with_empty_element semi-diff-closed cap-closed Subset-Family of X, P be pre-Measure of S holds ex M be Function of ( Ring_generated_by S), ExtREAL st (M . {} ) = 0 & for K be disjoint_valued FinSequence of S st ( Union K) in ( Ring_generated_by S) holds (M . ( Union K)) = ( Sum (P * K))

    proof

      let X be set, S be with_empty_element semi-diff-closed cap-closed Subset-Family of X, P be pre-Measure of S;

      consider M be nonnegative additive zeroed Function of ( Ring_generated_by S), ExtREAL such that

       A1: for A be set st A in ( Ring_generated_by S) holds for F be disjoint_valued FinSequence of S st A = ( Union F) holds (M . A) = ( Sum (P * F)) by Th46;

      take M;

      thus (M . {} ) = 0 by VALUED_0:def 19;

      thus for K be disjoint_valued FinSequence of S st ( Union K) in ( Ring_generated_by S) holds (M . ( Union K)) = ( Sum (P * K)) by A1;

    end;

    theorem :: MEASURE9:51

    for X,Z be set, P be with_empty_element semi-diff-closed cap-closed Subset-Family of X, K be disjoint_valued Function of NAT , ( Ring_generated_by P) st Z = { [n, F] where n be Nat, F be disjoint_valued FinSequence of P : ( Union F) = (K . n) & ((K . n) = {} implies F = <* {} *>) } holds ( proj2 Z) is FinSequenceSet of P & (for x be object holds x in ( rng K) iff ex F be FinSequence of P st F in ( proj2 Z) & ( Union F) = x) & ( proj2 Z) is with_non-empty_elements

    proof

      let X,Z be set, P be with_empty_element semi-diff-closed cap-closed Subset-Family of X, K be disjoint_valued Function of NAT , ( Ring_generated_by P);

      assume

       A1: Z = { [n, F] where n be Nat, F be disjoint_valued FinSequence of P : ( Union F) = (K . n) & ((K . n) = {} implies F = <* {} *>) };

      now

        let a be object;

        assume a in ( proj2 Z);

        then

        consider k be object such that

         A2: [k, a] in Z by XTUPLE_0:def 13;

        consider n be Nat, F be disjoint_valued FinSequence of P such that

         A3: [k, a] = [n, F] & ( Union F) = (K . n) & ((K . n) = {} implies F = <* {} *>) by A1, A2;

        thus a is FinSequence of P by A3, XTUPLE_0: 1;

      end;

      hence ( proj2 Z) is FinSequenceSet of P by FINSEQ_2:def 3;

      hereby

        let x be object;

        hereby

          assume x in ( rng K);

          then

          consider n be Element of NAT such that

           A6: x = (K . n) by FUNCT_2: 113;

          (K . n) in ( Ring_generated_by P);

          then (K . n) in ( DisUnion P) by SRINGS_3: 18;

          then

          consider A be Subset of X such that

           A7: x = A & ex F be disjoint_valued FinSequence of P st A = ( Union F) by A6;

          consider F be disjoint_valued FinSequence of P such that

           A8: A = ( Union F) by A7;

          per cases ;

            suppose

             A9: (K . n) = {} ;

            

             A10: ( rng <* {} *>) = { {} } by FINSEQ_1: 38;

             {} in P by SETFAM_1:def 8;

            then

            reconsider F1 = <* {} *> as disjoint_valued FinSequence of P by A10, ZFMISC_1: 31, FINSEQ_1:def 4;

            ( rng F1) = { {} } by FINSEQ_1: 38;

            then ( union ( rng F1)) = {} by ZFMISC_1: 25;

            then

             B1: ( Union F1) = {} by CARD_3:def 4;

            then [n, F1] in Z by A9, A1;

            then F1 in ( proj2 Z) by XTUPLE_0:def 13;

            hence ex F be FinSequence of P st F in ( proj2 Z) & ( Union F) = x by A9, B1, A6;

          end;

            suppose (K . n) <> {} ;

            then [n, F] in Z by A1, A6, A7, A8;

            then F in ( proj2 Z) by XTUPLE_0:def 13;

            hence ex F be FinSequence of P st F in ( proj2 Z) & ( Union F) = x by A8, A7;

          end;

        end;

        assume ex F be FinSequence of P st F in ( proj2 Z) & ( Union F) = x;

        then

        consider z be FinSequence of P such that

         A12: z in ( proj2 Z) & ( Union z) = x;

        consider y be object such that

         A13: [y, z] in Z by A12, XTUPLE_0:def 13;

        consider n be Nat, F be disjoint_valued FinSequence of P such that

         A14: [y, z] = [n, F] & ( Union F) = (K . n) & ((K . n) = {} implies F = <* {} *>) by A1, A13;

        y = n & z = F by A14, XTUPLE_0: 1;

        hence x in ( rng K) by A12, A14, FUNCT_2: 4, ORDINAL1:def 12;

      end;

      now

        assume {} in ( proj2 Z);

        then

        consider y be object such that

         A16: [y, {} ] in Z by XTUPLE_0:def 13;

        consider n be Nat, F be disjoint_valued FinSequence of P such that

         A17: [y, {} ] = [n, F] & ( Union F) = (K . n) & ((K . n) = {} implies F = <* {} *>) by A1, A16;

        y = n & {} = F by A17, XTUPLE_0: 1;

        then ( union ( rng F)) = {} by ZFMISC_1: 2;

        hence contradiction by A17, XTUPLE_0: 1, CARD_3:def 4;

      end;

      hence ( proj2 Z) is with_non-empty_elements;

    end;

    theorem :: MEASURE9:52

    for X be set, P be with_empty_element semi-diff-closed cap-closed Subset-Family of X, K be disjoint_valued Function of NAT , ( Ring_generated_by P) st ( rng K) is with_non-empty_element holds ex Y be non empty FinSequenceSet of P st Y = { F where F be disjoint_valued FinSequence of P : ( Union F) in ( rng K) & F <> {} } & Y is with_non-empty_elements

    proof

      let X be set, P be with_empty_element semi-diff-closed cap-closed Subset-Family of X, K be disjoint_valued Function of NAT , ( Ring_generated_by P);

      assume

       A0: ( rng K) is with_non-empty_element;

      set Y = { F where F be disjoint_valued FinSequence of P : ( Union F) in ( rng K) & F <> {} };

      now

        let a be object;

        assume a in Y;

        then ex A be disjoint_valued FinSequence of P st a = A & ( Union A) in ( rng K) & A <> {} ;

        hence a is FinSequence of P;

      end;

      then

      reconsider Y as FinSequenceSet of P by FINSEQ_2:def 3;

      consider k be non empty set such that

       A2: k in ( rng K) by A0;

      consider i be Element of NAT such that

       A3: k = (K . i) by A2, FUNCT_2: 113;

      (K . i) in ( Ring_generated_by P);

      then (K . i) in ( DisUnion P) by SRINGS_3: 18;

      then

      consider A be Subset of X such that

       A4: (K . i) = A & ex F be disjoint_valued FinSequence of P st A = ( Union F);

      consider F be disjoint_valued FinSequence of P such that

       A5: A = ( Union F) by A4;

      now

        assume F = {} ;

        then ( union ( rng F)) = {} by ZFMISC_1: 2;

        hence contradiction by A5, A4, A3, CARD_3:def 4;

      end;

      then F in Y by A2, A3, A4, A5;

      then

      reconsider Y as non empty FinSequenceSet of P;

      take Y;

      thus Y = { A where A be disjoint_valued FinSequence of P : ( Union A) in ( rng K) & A <> {} };

      now

        assume {} in Y;

        then ex A be disjoint_valued FinSequence of P st {} = A & ( Union A) in ( rng K) & A <> {} ;

        hence contradiction;

      end;

      hence Y is with_non-empty_elements;

    end;

    begin

    theorem :: MEASURE9:53

    

     Th51: for X,Z be set, P be semialgebra_of_sets of X, K be disjoint_valued Function of NAT , ( Field_generated_by P) st Z = { [n, F] where n be Nat, F be disjoint_valued FinSequence of P : ( Union F) = (K . n) & ((K . n) = {} implies F = <* {} *>) } holds ( proj2 Z) is FinSequenceSet of P & (for x be object holds x in ( rng K) iff ex F be FinSequence of P st F in ( proj2 Z) & ( Union F) = x) & ( proj2 Z) is with_non-empty_elements

    proof

      let X,Z be set, P be semialgebra_of_sets of X, K be disjoint_valued Function of NAT , ( Field_generated_by P);

      assume

       A1: Z = { [n, F] where n be Nat, F be disjoint_valued FinSequence of P : ( Union F) = (K . n) & ((K . n) = {} implies F = <* {} *>) };

      now

        let a be object;

        assume a in ( proj2 Z);

        then

        consider k be object such that

         A2: [k, a] in Z by XTUPLE_0:def 13;

        ex n be Nat, F be disjoint_valued FinSequence of P st [k, a] = [n, F] & ( Union F) = (K . n) & ((K . n) = {} implies F = <* {} *>) by A1, A2;

        hence a is FinSequence of P by XTUPLE_0: 1;

      end;

      hence ( proj2 Z) is FinSequenceSet of P by FINSEQ_2:def 3;

      hereby

        let x be object;

        hereby

          assume x in ( rng K);

          then

          consider n be Element of NAT such that

           A6: x = (K . n) by FUNCT_2: 113;

          (K . n) in ( Field_generated_by P);

          then (K . n) in ( DisUnion P) by SRINGS_3: 22;

          then

          consider A be Subset of X such that

           A7: x = A & ex F be disjoint_valued FinSequence of P st A = ( Union F) by A6;

          consider F be disjoint_valued FinSequence of P such that

           A8: A = ( Union F) by A7;

          per cases ;

            suppose

             A9: (K . n) = {} ;

            

             A10: ( rng <* {} *>) = { {} } by FINSEQ_1: 38;

             {} in P by SETFAM_1:def 8;

            then

            reconsider F1 = <* {} *> as disjoint_valued FinSequence of P by A10, ZFMISC_1: 31, FINSEQ_1:def 4;

            ( rng F1) = { {} } by FINSEQ_1: 38;

            then ( union ( rng F1)) = {} by ZFMISC_1: 25;

            then

             B1: ( Union F1) = {} by CARD_3:def 4;

            then [n, F1] in Z by A9, A1;

            then F1 in ( proj2 Z) by XTUPLE_0:def 13;

            hence ex F be FinSequence of P st F in ( proj2 Z) & ( Union F) = x by A9, B1, A6;

          end;

            suppose (K . n) <> {} ;

            then [n, F] in Z by A1, A6, A7, A8;

            then F in ( proj2 Z) by XTUPLE_0:def 13;

            hence ex F be FinSequence of P st F in ( proj2 Z) & ( Union F) = x by A8, A7;

          end;

        end;

        assume ex F be FinSequence of P st F in ( proj2 Z) & ( Union F) = x;

        then

        consider z be FinSequence of P such that

         A12: z in ( proj2 Z) & ( Union z) = x;

        consider y be object such that

         A13: [y, z] in Z by A12, XTUPLE_0:def 13;

        consider n be Nat, F be disjoint_valued FinSequence of P such that

         A14: [y, z] = [n, F] & ( Union F) = (K . n) & ((K . n) = {} implies F = <* {} *>) by A1, A13;

        y = n & z = F by A14, XTUPLE_0: 1;

        hence x in ( rng K) by A12, A14, ORDINAL1:def 12, FUNCT_2: 4;

      end;

      now

        assume {} in ( proj2 Z);

        then

        consider y be object such that

         A16: [y, {} ] in Z by XTUPLE_0:def 13;

        consider n be Nat, F be disjoint_valued FinSequence of P such that

         A17: [y, {} ] = [n, F] & ( Union F) = (K . n) & ((K . n) = {} implies F = <* {} *>) by A1, A16;

        y = n & {} = F by A17, XTUPLE_0: 1;

        then ( union ( rng F)) = {} by ZFMISC_1: 2;

        hence contradiction by A17, XTUPLE_0: 1, CARD_3:def 4;

      end;

      hence ( proj2 Z) is with_non-empty_elements;

    end;

    theorem :: MEASURE9:54

    

     Th54: for X be set, S be semialgebra_of_sets of X, P be pre-Measure of S, A be set holds for F1,F2 be disjoint_valued FinSequence of S st A = ( Union F1) & A = ( Union F2) holds ( Sum (P * F1)) = ( Sum (P * F2))

    proof

      let X be set, S be semialgebra_of_sets of X, P be pre-Measure of S, A be set;

      hereby

        let F1,F2 be disjoint_valued FinSequence of S;

        assume

         A1: A = ( Union F1) & A = ( Union F2);

        consider SMF1 be Function of NAT , ExtREAL such that

         A2: ( Sum (P * F1)) = (SMF1 . ( len (P * F1))) & (SMF1 . 0 ) = 0 & for i be Nat st i < ( len (P * F1)) holds (SMF1 . (i + 1)) = ((SMF1 . i) + ((P * F1) . (i + 1))) by EXTREAL1:def 2;

        consider SMF2 be Function of NAT , ExtREAL such that

         A3: ( Sum (P * F2)) = (SMF2 . ( len (P * F2))) & (SMF2 . 0 ) = 0 & for i be Nat st i < ( len (P * F2)) holds (SMF2 . (i + 1)) = ((SMF2 . i) + ((P * F2) . (i + 1))) by EXTREAL1:def 2;

        ( dom P) = S by FUNCT_2:def 1;

        then ( rng F1) c= ( dom P) & ( rng F2) c= ( dom P);

        then

         A4: ( dom (P * F1)) = ( dom F1) & ( dom (P * F2)) = ( dom F2) by RELAT_1: 27;

        then

         A5: ( dom (P * F1)) = ( Seg ( len F1)) & ( dom (P * F2)) = ( Seg ( len F2)) & ( len (P * F1)) = ( len F1) & ( len (P * F2)) = ( len F2) by FINSEQ_1:def 3, FINSEQ_3: 29;

        per cases ;

          suppose

           A6: ( len (P * F1)) = 0 ;

          then (P * F1) = {} ;

          then F1 = {} by A4;

          then ( rng F1) = {} ;

          then ( Union F2) = {} by A1, CARD_3:def 4, ZFMISC_1: 2;

          then

           A7: ( union ( rng F2)) = {} by CARD_3:def 4;

          defpred S[ Nat] means $1 <= ( len (P * F2)) implies (SMF2 . $1) = 0 ;

          

           A8: S[ 0 ] by A3;

          

           A9: for i be Nat st S[i] holds S[(i + 1)]

          proof

            let i be Nat;

            assume

             A10: S[i];

            assume

             A11: (i + 1) <= ( len (P * F2));

            then

             A13: (SMF2 . (i + 1)) = ((SMF2 . i) + ((P * F2) . (i + 1))) & (SMF2 . i) = 0 by A3, A10, NAT_1: 13;

            

             A14: (i + 1) in ( dom (P * F2)) by A11, NAT_1: 11, FINSEQ_3: 25;

            then (F2 . (i + 1)) = {} by A4, A7, ORDERS_1: 6, FUNCT_1: 3;

            then (P . (F2 . (i + 1))) = 0 by VALUED_0:def 19;

            then ((P * F2) . (i + 1)) = 0 by A14, FUNCT_1: 12;

            hence (SMF2 . (i + 1)) = 0 by A13;

          end;

          for i be Nat holds S[i] from NAT_1:sch 2( A8, A9);

          hence ( Sum (P * F1)) = ( Sum (P * F2)) by A2, A3, A6;

        end;

          suppose

           B6: ( len (P * F2)) = 0 ;

          then (P * F2) = {} ;

          then F2 = {} by A4;

          then ( rng F2) = {} ;

          then ( Union F1) = {} by A1, CARD_3:def 4, ZFMISC_1: 2;

          then

           B7: ( union ( rng F1)) = {} by CARD_3:def 4;

          defpred S[ Nat] means $1 <= ( len (P * F1)) implies (SMF1 . $1) = 0 ;

          

           B8: S[ 0 ] by A2;

          

           B9: for i be Nat st S[i] holds S[(i + 1)]

          proof

            let i be Nat;

            assume

             B10: S[i];

            assume

             B11: (i + 1) <= ( len (P * F1));

            then

             B13: (SMF1 . (i + 1)) = ((SMF1 . i) + ((P * F1) . (i + 1))) & (SMF1 . i) = 0 by A2, B10, NAT_1: 13;

            

             B14: (i + 1) in ( dom (P * F1)) by B11, NAT_1: 11, FINSEQ_3: 25;

            then (F1 . (i + 1)) = {} by A4, B7, ORDERS_1: 6, FUNCT_1: 3;

            then (P . (F1 . (i + 1))) = 0 by VALUED_0:def 19;

            then ((P * F1) . (i + 1)) = 0 by B14, FUNCT_1: 12;

            hence (SMF1 . (i + 1)) = 0 by B13;

          end;

          for i be Nat holds S[i] from NAT_1:sch 2( B8, B9);

          hence ( Sum (P * F1)) = ( Sum (P * F2)) by A2, A3, B6;

        end;

          suppose

           A15: ( len (P * F1)) <> 0 & ( len (P * F2)) <> 0 ;

          defpred Mx[ Nat, Nat, set] means $3 = (P . ((F1 . $1) /\ (F2 . $2)));

          

           MX0: for i,j be Nat st [i, j] in [:( Seg ( len F1)), ( Seg ( len F2)):] holds ex A be Element of ExtREAL st Mx[i, j, A];

          consider Mx be Matrix of ( len F1), ( len F2), ExtREAL such that

           MX1: for i,j be Nat st [i, j] in ( Indices Mx) holds Mx[i, j, (Mx * (i,j))] from MATRIX_0:sch 2( MX0);

          

           C3: ( len Mx) = ( len F1) by MATRIX_0:def 2;

          then

           C4: ( width Mx) = ( len F2) by A15, A5, MATRIX_0: 20;

          

           CC0: for F be disjoint_valued FinSequence of S st ( Union F) in S holds (P . ( Union F)) = ( Sum (P * F)) by Def8;

          F1 is non empty & F2 is non empty by A15;

          then

           C10: ( Sum (P * F1)) = ( SumAll Mx) & ( Sum (P * F2)) = ( SumAll (Mx @ )) by A1, MX1, CC0, Th40, Th41;

          for i be Nat st i in ( dom Mx) holds not -infty in ( rng (Mx . i))

          proof

            let i be Nat;

            assume

             F1: i in ( dom Mx);

            assume -infty in ( rng (Mx . i));

            then

            consider j be object such that

             F2: j in ( dom (Mx . i)) & ((Mx . i) . j) = -infty by FUNCT_1:def 3;

            reconsider j as Nat by F2;

            

             F3: [i, j] in ( Indices Mx) by F1, F2, MATRPROB: 13;

            then ((Mx . i) . j) = (Mx * (i,j)) by MATRPROB: 14;

            then

             F5: ((Mx . i) . j) = (P . ((F1 . i) /\ (F2 . j))) by F3, MX1;

            i in ( Seg ( len Mx)) & j in ( Seg ( width Mx)) by F3, MATRPROB: 12;

            then i in ( dom F1) & j in ( dom F2) by C3, C4, FINSEQ_1:def 3;

            then (F1 . i) in ( rng F1) & (F2 . j) in ( rng F2) by FUNCT_1: 3;

            then ((F1 . i) /\ (F2 . j)) in S by FINSUB_1:def 2;

            hence contradiction by F2, F5, MEASURE1:def 2;

          end;

          hence ( Sum (P * F1)) = ( Sum (P * F2)) by C10, Th28;

        end;

      end;

    end;

    theorem :: MEASURE9:55

    

     Th55: for X be set, S be semialgebra_of_sets of X, P be pre-Measure of S holds ex M be Measure of ( Field_generated_by S) st for A be set st A in ( Field_generated_by S) holds for F be disjoint_valued FinSequence of S st A = ( Union F) holds (M . A) = ( Sum (P * F))

    proof

      let X be set, S be semialgebra_of_sets of X, P be pre-Measure of S;

      defpred P[ object, object] means for F be disjoint_valued FinSequence of S st $1 = ( Union F) holds $2 = ( Sum (P * F));

      

       A1: for A be object st A in ( Field_generated_by S) holds ex p be object st p in ExtREAL & P[A, p]

      proof

        let A be object;

        assume A in ( Field_generated_by S);

        then A in ( DisUnion S) by SRINGS_3: 22;

        then

        consider V be Subset of X such that

         A3: A = V & ex F be disjoint_valued FinSequence of S st V = ( Union F);

        consider F be disjoint_valued FinSequence of S such that

         A4: V = ( Union F) by A3;

        set p = ( Sum (P * F));

        take p;

        thus p in ExtREAL & P[A, p] by A3, A4, Th54;

      end;

      consider M be Function of ( Field_generated_by S), ExtREAL such that

       A5: for A be object st A in ( Field_generated_by S) holds P[A, (M . A)] from FUNCT_2:sch 1( A1);

      

       A18: for A be Element of ( Field_generated_by S) holds 0 <= (M . A)

      proof

        let A be Element of ( Field_generated_by S);

        A in ( Field_generated_by S);

        then A in ( DisUnion S) by SRINGS_3: 22;

        then

        consider V be Subset of X such that

         A7: A = V & ex F be disjoint_valued FinSequence of S st V = ( Union F);

        consider F be disjoint_valued FinSequence of S such that

         A8: V = ( Union F) by A7;

        consider PF be sequence of ExtREAL such that

         A10: ( Sum (P * F)) = (PF . ( len (P * F))) & (PF . 0 ) = 0. & for i be Nat st i < ( len (P * F)) holds (PF . (i + 1)) = ((PF . i) + ((P * F) . (i + 1))) by EXTREAL1:def 2;

        defpred P2[ Nat] means $1 <= ( len (P * F)) implies (PF . $1) >= 0 ;

        

         A11: P2[ 0 ] by A10;

        

         A12: for i be Nat st P2[i] holds P2[(i + 1)]

        proof

          let i be Nat;

          assume

           A13: P2[i];

          assume

           A14: (i + 1) <= ( len (P * F));

          then (i + 1) in ( dom (P * F)) by NAT_1: 11, FINSEQ_3: 25;

          then ((P * F) . (i + 1)) = (P . (F . (i + 1))) by FUNCT_1: 12;

          then

           A17: ((P * F) . (i + 1)) >= 0 by SUPINF_2: 51;

          (PF . (i + 1)) = ((PF . i) + ((P * F) . (i + 1))) by A14, A10, NAT_1: 13;

          hence (PF . (i + 1)) >= 0 by A13, A14, NAT_1: 13, A17;

        end;

        for i be Nat holds P2[i] from NAT_1:sch 2( A11, A12);

        then ( Sum (P * F)) >= 0 by A10;

        hence 0 <= (M . A) by A7, A8, A5;

      end;

      

       A29: for A,B be Element of ( Field_generated_by S) st A misses B holds (M . (A \/ B)) = ((M . A) + (M . B))

      proof

        let A,B be Element of ( Field_generated_by S);

        assume

         A19: A misses B;

        A in ( Field_generated_by S);

        then A in ( DisUnion S) by SRINGS_3: 22;

        then

        consider V be Subset of X such that

         A20: A = V & ex F be disjoint_valued FinSequence of S st V = ( Union F);

        consider F be disjoint_valued FinSequence of S such that

         A21: V = ( Union F) by A20;

        B in ( Field_generated_by S);

        then B in ( DisUnion S) by SRINGS_3: 22;

        then

        consider W be Subset of X such that

         A22: B = W & ex G be disjoint_valued FinSequence of S st W = ( Union G);

        consider G be disjoint_valued FinSequence of S such that

         A23: W = ( Union G) by A22;

        set H = (F ^ G);

        

         A24: A = ( union ( rng F)) & B = ( union ( rng G)) by A20, A21, A22, A23, CARD_3:def 4;

        then

        reconsider H as disjoint_valued FinSequence of S by A19, Th43;

        ( rng H) = (( rng F) \/ ( rng G)) by FINSEQ_1: 31;

        then ( union ( rng H)) = (( union ( rng F)) \/ ( union ( rng G))) by ZFMISC_1: 78;

        then (A \/ B) = ( Union H) by A24, CARD_3:def 4;

        then

         A25: (M . (A \/ B)) = ( Sum (P * H)) by A5;

        

         A26: (M . A) = ( Sum (P * F)) & (M . B) = ( Sum (P * G)) by A20, A21, A22, A23, A5;

        (P * F) is nonnegative by Th45;

        then

         A27: not -infty in ( rng (P * F)) by SUPINF_2:def 12, SUPINF_2:def 9;

        (P * G) is nonnegative by Th45;

        then

         A28: not -infty in ( rng (P * G)) by SUPINF_2:def 12, SUPINF_2:def 9;

        (P * H) = ((P * F) ^ (P * G)) by FINSEQOP: 9;

        hence (M . (A \/ B)) = ((M . A) + (M . B)) by A25, A26, A27, A28, EXTREAL1: 10;

      end;

      reconsider E = {} as Element of S by SETFAM_1:def 8;

      reconsider F = <*E*> as disjoint_valued FinSequence of S;

      ( rng F) = { {} } by FINSEQ_1: 38;

      then ( union ( rng F)) = {} by ZFMISC_1: 25;

      then ( Union F) = {} by CARD_3:def 4;

      then (M . {} ) = ( Sum (P * F)) by A5, FINSUB_1: 7;

      then (M . {} ) = ( Sum <*(P . {} )*>) by FINSEQ_2: 35;

      then (M . {} ) = (P . {} ) by EXTREAL1: 8;

      then (M . {} ) = 0 by VALUED_0:def 19;

      then

      reconsider M as nonnegative additive zeroed Function of ( Field_generated_by S), ExtREAL by A18, A29, VALUED_0:def 19, MEASURE1:def 2, MEASURE1:def 8;

      take M;

      thus thesis by A5;

    end;

    theorem :: MEASURE9:56

    for F be ExtREAL_sequence, n be Nat, a be R_eal st (for k be Nat holds (F . k) = a) holds (( Partial_Sums F) . n) = (a * (n + 1))

    proof

      let F be ExtREAL_sequence, n be Nat, a be R_eal;

      assume

       A1: for k be Nat holds (F . k) = a;

      defpred P[ Nat] means (( Partial_Sums F) . $1) = (a * ($1 + 1));

      (( Partial_Sums F) . 0 ) = (F . 0 ) by MESFUNC9:def 1;

      then (( Partial_Sums F) . 0 ) = a by A1;

      then

       A2: P[ 0 ] by XXREAL_3: 81;

      

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

      proof

        let i be Nat;

        assume

         A4: P[i];

        (i + 1) in REAL & 1 in REAL by XREAL_0:def 1;

        then

        reconsider i1 = (i + 1), One = 1 as R_eal by XBOOLE_0:def 3, XXREAL_0:def 4;

        (( Partial_Sums F) . (i + 1)) = ((( Partial_Sums F) . i) + (F . (i + 1))) by MESFUNC9:def 1;

        then (( Partial_Sums F) . (i + 1)) = ((a * (i + 1)) + a) by A1, A4;

        then (( Partial_Sums F) . (i + 1)) = ((a * (i + 1)) + (a * 1)) by XXREAL_3: 81;

        then (( Partial_Sums F) . (i + 1)) = (a * (i1 + One)) by XXREAL_3: 96;

        hence P[(i + 1)] by XXREAL_3:def 2;

      end;

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

      hence thesis;

    end;

    theorem :: MEASURE9:57

    

     Th57: for X be non empty set, F be sequence of X, n be Nat holds ( rng (F | ( Segm (n + 1)))) = (( rng (F | ( Segm n))) \/ {(F . n)})

    proof

      let X be non empty set, F be sequence of X, n be Nat;

      now

        let y be object;

        assume y in ( rng (F | ( Segm (n + 1))));

        then

        consider x be object such that

         A1: x in ( dom (F | ( Segm (n + 1)))) & y = ((F | ( Segm (n + 1))) . x) by FUNCT_1:def 3;

        reconsider x as Nat by A1;

        

         A4: y = (F . x) by A1, FUNCT_1: 47;

        x in (( dom F) /\ ( Segm (n + 1))) by A1, RELAT_1: 61;

        then

         A2: x in ( dom F) & x in ( Segm (n + 1)) by XBOOLE_0:def 4;

        x < (n + 1) by A2, NAT_1: 44;

        then

         A3: x <= n by NAT_1: 13;

        per cases ;

          suppose x = n;

          then y in {(F . n)} by A4, TARSKI:def 1;

          hence y in (( rng (F | ( Segm n))) \/ {(F . n)}) by XBOOLE_0:def 3;

        end;

          suppose x <> n;

          then x < n by A3, XXREAL_0: 1;

          then x in ( Segm n) by NAT_1: 44;

          then x in (( dom F) /\ ( Segm n)) by A2, XBOOLE_0:def 4;

          then x in ( dom (F | ( Segm n))) by RELAT_1: 61;

          then ((F | ( Segm n)) . x) in ( rng (F | ( Segm n))) & ((F | ( Segm n)) . x) = (F . x) by FUNCT_1: 3, FUNCT_1: 47;

          hence y in (( rng (F | ( Segm n))) \/ {(F . n)}) by A4, XBOOLE_0:def 3;

        end;

      end;

      then

       A5: ( rng (F | ( Segm (n + 1)))) c= (( rng (F | ( Segm n))) \/ {(F . n)}) by TARSKI:def 3;

      now

        let y be object;

        assume

         A6: y in (( rng (F | ( Segm n))) \/ {(F . n)});

        per cases by A6, XBOOLE_0:def 3;

          suppose

           A7: y in ( rng (F | ( Segm n)));

          n <= (n + 1) by NAT_1: 11;

          then (F | ( Segm n)) c= (F | ( Segm (n + 1))) by NAT_1: 39, RELAT_1: 75;

          then ( rng (F | ( Segm n))) c= ( rng (F | ( Segm (n + 1)))) by RELAT_1: 11;

          hence y in ( rng (F | ( Segm (n + 1)))) by A7;

        end;

          suppose y in {(F . n)};

          then

           A8: y = (F . n) by TARSKI:def 1;

          n in NAT by ORDINAL1:def 12;

          then n in ( dom F) & n in ( Segm (n + 1)) by FUNCT_2:def 1, NAT_1: 45;

          then n in (( dom F) /\ ( Segm (n + 1))) by XBOOLE_0:def 4;

          then

           A9: n in ( dom (F | ( Segm (n + 1)))) by RELAT_1: 61;

          then (F . n) = ((F | ( Segm (n + 1))) . n) by FUNCT_1: 47;

          hence y in ( rng (F | ( Segm (n + 1)))) by A8, A9, FUNCT_1: 3;

        end;

      end;

      then (( rng (F | ( Segm n))) \/ {(F . n)}) c= ( rng (F | ( Segm (n + 1)))) by TARSKI:def 3;

      hence thesis by A5, XBOOLE_0:def 10;

    end;

    theorem :: MEASURE9:58

    

     Th58: for X be set, S be Field_Subset of X, M be Measure of S, F be Sep_Sequence of S, n be Nat holds ( union ( rng (F | ( Segm (n + 1))))) in S & (( Partial_Sums (M * F)) . n) = (M . ( union ( rng (F | ( Segm (n + 1))))))

    proof

      let X be set, S be Field_Subset of X, M be Measure of S, F be Sep_Sequence of S, n be Nat;

      

       A2: ( rng (F | ( Segm ( 0 + 1)))) = (( rng (F | ( Segm 0 ))) \/ {(F . 0 )}) by Th57

      .= {(F . 0 )};

      then

       A2a: ( union ( rng (F | ( Segm ( 0 + 1))))) = (F . 0 ) by ZFMISC_1: 25;

      defpred P2[ Nat] means ( union ( rng (F | ( Segm ($1 + 1))))) in S;

      

       A14: P2[ 0 ] by A2a;

      

       A15: for k be Nat st P2[k] holds P2[(k + 1)]

      proof

        let k be Nat;

        assume

         A16: P2[k];

        ( union ( rng (F | ( Segm ((k + 1) + 1))))) = ( union (( rng (F | ( Segm (k + 1)))) \/ {(F . (k + 1))})) by Th57

        .= (( union ( rng (F | ( Segm (k + 1))))) \/ ( union {(F . (k + 1))})) by ZFMISC_1: 78

        .= (( union ( rng (F | ( Segm (k + 1))))) \/ (F . (k + 1))) by ZFMISC_1: 25;

        hence ( union ( rng (F | ( Segm ((k + 1) + 1))))) in S by A16, PROB_1: 3;

      end;

      

       P1: for k be Nat holds P2[k] from NAT_1:sch 2( A14, A15);

      hence ( union ( rng (F | ( Segm (n + 1))))) in S;

      defpred P[ Nat] means (( Partial_Sums (M * F)) . $1) = (M . ( union ( rng (F | ( Segm ($1 + 1))))));

      

       A1: (( Partial_Sums (M * F)) . 0 ) = ((M * F) . 0 ) by MESFUNC9:def 1

      .= (M . (F . 0 )) by FUNCT_2: 15;

      

       A3: P[ 0 ] by A1, A2, ZFMISC_1: 25;

      

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

      proof

        let n be Nat;

        assume

         A5: P[n];

        

         A6: (( Partial_Sums (M * F)) . (n + 1)) = ((( Partial_Sums (M * F)) . n) + ((M * F) . (n + 1))) by MESFUNC9:def 1

        .= ((M . ( union ( rng (F | ( Segm (n + 1)))))) + (M . (F . (n + 1)))) by A5, FUNCT_2: 15;

         A13:

        now

          assume ex x be object st x in (( union ( rng (F | ( Segm (n + 1))))) /\ (F . (n + 1)));

          then

          consider x be object such that

           A7: x in (( union ( rng (F | ( Segm (n + 1))))) /\ (F . (n + 1)));

          

           A8: x in ( union ( rng (F | ( Segm (n + 1))))) & x in (F . (n + 1)) by A7, XBOOLE_0:def 4;

          then

          consider A be set such that

           A9: x in A & A in ( rng (F | ( Segm (n + 1)))) by TARSKI:def 4;

          consider k be object such that

           A10: k in ( dom (F | ( Segm (n + 1)))) & A = ((F | ( Segm (n + 1))) . k) by A9, FUNCT_1:def 3;

          reconsider k as Nat by A10;

          

           A11: k < (n + 1) by A10, RELAT_1: 57, NAT_1: 44;

          A = (F . k) by A10, FUNCT_1: 47;

          then x in ((F . k) /\ (F . (n + 1))) by A8, A9, XBOOLE_0:def 4;

          hence contradiction by A11, PROB_2:def 2, XBOOLE_0: 4;

        end;

        ( union ( rng (F | ( Segm (n + 1))))) in S by P1;

        

        then ((M . ( union ( rng (F | ( Segm (n + 1)))))) + (M . (F . (n + 1)))) = (M . (( union ( rng (F | ( Segm (n + 1))))) \/ (F . (n + 1)))) by A13, XBOOLE_0: 4, MEASURE1:def 8

        .= (M . (( union ( rng (F | ( Segm (n + 1))))) \/ ( union {(F . (n + 1))}))) by ZFMISC_1: 25

        .= (M . ( union (( rng (F | ( Segm (n + 1)))) \/ {(F . (n + 1))}))) by ZFMISC_1: 78

        .= (M . ( union ( rng (F | ( Segm ((n + 1) + 1)))))) by Th57;

        hence thesis by A6;

      end;

      for n be Nat holds P[n] from NAT_1:sch 2( A3, A4);

      hence (( Partial_Sums (M * F)) . n) = (M . ( union ( rng (F | ( Segm (n + 1))))));

    end;

    theorem :: MEASURE9:59

    

     Th59: for X be set, S be semialgebra_of_sets of X, P be pre-Measure of S, M be Measure of ( Field_generated_by S) st (for A be set st A in ( Field_generated_by S) holds for F be disjoint_valued FinSequence of S st A = ( Union F) holds (M . A) = ( Sum (P * F))) holds M is completely-additive

    proof

      let X be set, S be semialgebra_of_sets of X, P be pre-Measure of S, M be Measure of ( Field_generated_by S);

      assume

       A1: for A be set st A in ( Field_generated_by S) holds for F be disjoint_valued FinSequence of S st A = ( Union F) holds (M . A) = ( Sum (P * F));

      now

        let FSets be Sep_Sequence of ( Field_generated_by S);

        assume

         B0: ( union ( rng FSets)) in ( Field_generated_by S);

        then ( union ( rng FSets)) in ( DisUnion S) by SRINGS_3: 22;

        then

        consider A be Subset of X such that

         B1: A = ( union ( rng FSets)) & ex F be disjoint_valued FinSequence of S st A = ( Union F);

        consider D be disjoint_valued FinSequence of S such that

         B2: A = ( Union D) by B1;

        set Z = { [n, E] where n be Nat, E be disjoint_valued FinSequence of S : ( Union E) = (FSets . n) & ((FSets . n) = {} implies E = <* {} *>) };

        reconsider Y = ( proj2 Z) as FinSequenceSet of S by Th51;

        

         E4: Y is with_non-empty_elements by Th51;

        per cases ;

          suppose ( rng FSets) is with_non-empty_element;

          then

          consider a be non empty set such that

           E6: a in ( rng FSets);

          ex E be FinSequence of S st E in Y & ( Union E) = a by E6, Th51;

          then

          reconsider Y as non empty with_non-empty_element FinSequenceSet of S by E4;

          defpred P[ Element of NAT , object] means [$1, $2] in Z;

          

           F2: for n be Element of NAT holds ex y be Element of Y st P[n, y]

          proof

            let n be Element of NAT ;

            (FSets . n) in ( Field_generated_by S);

            then (FSets . n) in ( DisUnion S) by SRINGS_3: 22;

            then

            consider A be Subset of X such that

             F3: (FSets . n) = A & ex F be disjoint_valued FinSequence of S st A = ( Union F);

            consider F be disjoint_valued FinSequence of S such that

             F4: A = ( Union F) by F3;

            per cases ;

              suppose

               F5: (FSets . n) = {} ;

              

               F6: ( rng <* {} *>) = { {} } by FINSEQ_1: 38;

               {} in S by SETFAM_1:def 8;

              then

              reconsider E = <* {} *> as disjoint_valued FinSequence of S by F6, ZFMISC_1: 31, FINSEQ_1:def 4;

              ( union ( rng E)) = {} by F6, ZFMISC_1: 25;

              then ( Union E) = {} by CARD_3:def 4;

              then

               F7: [n, E] in Z by F5;

              then E in Y by XTUPLE_0:def 13;

              hence ex y be Element of Y st P[n, y] by F7;

            end;

              suppose (FSets . n) <> {} ;

              then

               F8: [n, F] in Z by F4, F3;

              then F in Y by XTUPLE_0:def 13;

              hence ex y be Element of Y st P[n, y] by F8;

            end;

          end;

          consider s be Function of NAT , Y such that

           F9: for n be Element of NAT holds P[n, (s . n)] from FUNCT_2:sch 3( F2);

          now

            let n be object;

            assume n in ( dom s);

            then

            reconsider n1 = n as Element of NAT ;

             [n1, (s . n1)] in Z by F9;

            then

             F11: ex m be Nat, E be disjoint_valued FinSequence of S st [n1, (s . n1)] = [m, E] & ( Union E) = (FSets . m) & ((FSets . m) = {} implies E = <* {} *>);

            now

              assume

               F15: (s . n) = {} ;

              then ( union ( rng (s . n1))) = {} by ZFMISC_1: 2;

              then ( Union (s . n1)) = {} by CARD_3:def 4;

              hence contradiction by F11, F15, XTUPLE_0: 1;

            end;

            hence (s . n) is non empty;

          end;

          then

          reconsider s as non-empty sequence of Y by FUNCT_1:def 9;

          reconsider G = ( joined_seq s) as sequence of S;

          now

            let x,y be object;

            assume

             F16: x <> y;

            per cases ;

              suppose not x in NAT or not y in NAT ;

              then not x in ( dom G) or not y in ( dom G);

              then (G . x) = {} or (G . y) = {} by FUNCT_1:def 2;

              hence (G . x) misses (G . y) by XBOOLE_1: 65;

            end;

              suppose x in NAT & y in NAT ;

              then

              reconsider n1 = x, n2 = y as Element of NAT ;

              consider k1,m1 be Nat such that

               F17: m1 in ( dom (s . k1)) & ((((( Partial_Sums ( Length s)) . k1) - ( len (s . k1))) + m1) - 1) = n1 & (G . n1) = ((s . k1) . m1) by Def4;

              consider k2,m2 be Nat such that

               F18: m2 in ( dom (s . k2)) & ((((( Partial_Sums ( Length s)) . k2) - ( len (s . k2))) + m2) - 1) = n2 & (G . n2) = ((s . k2) . m2) by Def4;

              k1 is Element of NAT & k2 is Element of NAT by ORDINAL1:def 12;

              then

               F21: [k1, (s . k1)] in Z & [k2, (s . k2)] in Z by F9;

              then

              consider i1 be Nat, E1 be disjoint_valued FinSequence of S such that

               F22: [k1, (s . k1)] = [i1, E1] & ( Union E1) = (FSets . i1) & ((FSets . i1) = {} implies E1 = <* {} *>);

              consider i2 be Nat, E2 be disjoint_valued FinSequence of S such that

               F23: [k2, (s . k2)] = [i2, E2] & ( Union E2) = (FSets . i2) & ((FSets . i2) = {} implies E2 = <* {} *>) by F21;

              

               F24: k1 = i1 & (s . k1) = E1 & k2 = i2 & (s . k2) = E2 by F22, F23, XTUPLE_0: 1;

              now

                assume k1 <> k2;

                then (FSets . i1) misses (FSets . i2) by F24, PROB_2:def 2;

                then ( union ( rng (s . k1))) misses ( Union (s . k2)) by F22, F23, F24, CARD_3:def 4;

                then

                 F25: ( union ( rng (s . k1))) misses ( union ( rng (s . k2))) by CARD_3:def 4;

                (G . n1) c= ( union ( rng (s . k1))) & (G . n2) c= ( union ( rng (s . k2))) by F17, F18, FUNCT_1: 3, ZFMISC_1: 74;

                hence (G . n1) misses (G . n2) by F25, XBOOLE_1: 64;

              end;

              hence (G . x) misses (G . y) by F16, F17, F18, F24, PROB_2:def 2;

            end;

          end;

          then

          reconsider G as disjoint_valued sequence of S by PROB_2:def 2;

          now

            let x be object;

            assume x in ( Union FSets);

            then x in ( union ( rng FSets)) by CARD_3:def 4;

            then

            consider A be set such that

             G1: x in A & A in ( rng FSets) by TARSKI:def 4;

            consider n be Element of NAT such that

             G2: A = (FSets . n) by G1, FUNCT_2: 113;

             [n, (s . n)] in Z by F9;

            then

            consider n2 be Nat, E2 be disjoint_valued FinSequence of S such that

             G6: [n, (s . n)] = [n2, E2] & ( Union E2) = (FSets . n2) & ((FSets . n2) = {} implies E2 = <* {} *>);

            n = n2 & (s . n) = E2 by G6, XTUPLE_0: 1;

            then x in ( union ( rng (s . n))) by G1, G2, G6, CARD_3:def 4;

            then

            consider A2 be set such that

             G8: x in A2 & A2 in ( rng (s . n)) by TARSKI:def 4;

            consider m be object such that

             G9: m in ( dom (s . n)) & A2 = ((s . n) . m) by G8, FUNCT_1:def 3;

            reconsider m as Nat by G9;

            consider N be Nat such that

             G10: N = ((((( Partial_Sums ( Length s)) . n) - ( len (s . n))) + m) - 1) & (G . N) = ((s . n) . m) by G9, Th13;

            A2 in ( rng G) by FUNCT_2: 4, G9, G10, ORDINAL1:def 12;

            then x in ( union ( rng G)) by G8, TARSKI:def 4;

            hence x in ( Union G) by CARD_3:def 4;

          end;

          then

           T0: ( Union FSets) c= ( Union G) by TARSKI:def 3;

          now

            let x be object;

            assume x in ( Union G);

            then x in ( union ( rng G)) by CARD_3:def 4;

            then

            consider A be set such that

             G11: x in A & A in ( rng G) by TARSKI:def 4;

            consider n be Element of NAT such that

             G12: A = (G . n) by G11, FUNCT_2: 113;

            consider k,m be Nat such that

             G13: m in ( dom (s . k)) & ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) = n & (G . n) = ((s . k) . m) by Def4;

            k is Element of NAT by ORDINAL1:def 12;

            then [k, (s . k)] in Z by F9;

            then

            consider k2 be Nat, E be disjoint_valued FinSequence of S such that

             G14: [k, (s . k)] = [k2, E] & ( Union E) = (FSets . k2) & ((FSets . k2) = {} implies E = <* {} *>);

            

             G15: k = k2 & (s . k) = E by G14, XTUPLE_0: 1;

            x in ((s . k) . m) & ((s . k) . m) in ( rng (s . k)) by G11, G12, G13, FUNCT_1: 3;

            then x in ( union ( rng (s . k))) by TARSKI:def 4;

            then

             G16: x in (FSets . k2) by G14, G15, CARD_3:def 4;

            (FSets . k2) in ( rng FSets) by FUNCT_2: 4, ORDINAL1:def 12;

            then x in ( union ( rng FSets)) by G16, TARSKI:def 4;

            hence x in ( Union FSets) by CARD_3:def 4;

          end;

          then ( Union G) c= ( Union FSets) by TARSKI:def 3;

          then

           T1: ( Union FSets) = ( Union G) by T0, XBOOLE_0:def 10;

          defpred Q[ Nat, Nat, object] means (($1 + 1) <= ( len D) implies $3 = ((D . ($1 + 1)) /\ (G . $2))) & (($1 + 1) > ( len D) implies $3 = {} );

          

           D0: for i be Element of NAT holds for j be Element of NAT holds ex z be Element of S st Q[i, j, z]

          proof

            let i,j be Element of NAT ;

            per cases ;

              suppose

               D1: (i + 1) <= ( len D);

              then (i + 1) in ( dom D) by NAT_1: 11, FINSEQ_3: 25;

              then (D . (i + 1)) in ( rng D) by FUNCT_1: 3;

              then ((D . (i + 1)) /\ (G . j)) in S by FINSUB_1:def 2;

              hence ex z be Element of S st Q[i, j, z] by D1;

            end;

              suppose

               D4: (i + 1) > ( len D);

               {} in S by SETFAM_1:def 8;

              hence ex z be Element of S st Q[i, j, z] by D4;

            end;

          end;

          consider LG be Function of [: NAT , NAT :], S such that

           D5: for i be Element of NAT holds for j be Element of NAT holds Q[i, j, (LG . (i,j))] from BINOP_1:sch 3( D0);

          

           D5a: for i,j be Nat holds ((i + 1) <= ( len D) implies (LG . (i,j)) = ((D . (i + 1)) /\ (G . j))) & ((i + 1) > ( len D) implies (LG . (i,j)) = {} )

          proof

            let i,j be Nat;

            reconsider i1 = i, j1 = j as Element of NAT by ORDINAL1:def 12;

             DD5:

            now

              assume (i + 1) <= ( len D);

              then (LG . (i1,j1)) = ((D . (i + 1)) /\ (G . j)) by D5;

              hence (LG . (i,j)) = ((D . (i + 1)) /\ (G . j));

            end;

            now

              assume (i + 1) > ( len D);

              then (LG . (i1,j1)) = {} by D5;

              hence (LG . (i,j)) = {} ;

            end;

            hence thesis by DD5;

          end;

          ( Union FSets) = ( union ( rng FSets)) by CARD_3:def 4;

          then

           X2: (M . ( Union FSets)) = ( Sum (P * D)) by A1, B0, B1, B2;

          consider SumPD be sequence of ExtREAL such that

           X3: ( Sum (P * D)) = (SumPD . ( len (P * D))) & (SumPD . 0 ) = 0. & for i be Nat st i < ( len (P * D)) holds (SumPD . (i + 1)) = ((SumPD . i) + ((P * D) . (i + 1))) by EXTREAL1:def 2;

          

           X4: for i be Element of NAT st i < ( len D) holds (D . (i + 1)) = ( Union ( ProjMap1 (LG,i)))

          proof

            let i be Element of NAT ;

            assume

             X40: i < ( len D);

            then 1 <= (i + 1) & (i + 1) <= ( len D) by NAT_1: 11, NAT_1: 13;

            then (i + 1) in ( dom D) by FINSEQ_3: 25;

            then

             X41: (D . (i + 1)) in ( rng D) by FUNCT_1: 3;

            now

              let x be object;

              assume

               X44: x in (D . (i + 1));

              then x in ( union ( rng D)) by X41, TARSKI:def 4;

              then x in ( Union D) by CARD_3:def 4;

              then x in ( Union G) by B1, B2, T1, CARD_3:def 4;

              then x in ( union ( rng G)) by CARD_3:def 4;

              then

              consider Gx be set such that

               X42: x in Gx & Gx in ( rng G) by TARSKI:def 4;

              consider j be Element of NAT such that

               X43: Gx = (G . j) by X42, FUNCT_2: 113;

              

               X46: ( dom ( ProjMap1 (LG,i))) = NAT by FUNCT_2:def 1;

              

               X45: x in ((D . (i + 1)) /\ (G . j)) by X44, X42, X43, XBOOLE_0:def 4;

              (i + 1) <= ( len D) implies (LG . (i,j)) = ((D . (i + 1)) /\ (G . j)) by D5;

              then

               X47: x in (( ProjMap1 (LG,i)) . j) by X40, NAT_1: 13, X45, MESFUNC9:def 6;

              (( ProjMap1 (LG,i)) . j) in ( rng ( ProjMap1 (LG,i))) by X46, FUNCT_1: 3;

              then x in ( union ( rng ( ProjMap1 (LG,i)))) by X47, TARSKI:def 4;

              hence x in ( Union ( ProjMap1 (LG,i))) by CARD_3:def 4;

            end;

            then

             X48: (D . (i + 1)) c= ( Union ( ProjMap1 (LG,i))) by TARSKI:def 3;

            now

              let x be object;

              assume x in ( Union ( ProjMap1 (LG,i)));

              then x in ( union ( rng ( ProjMap1 (LG,i)))) by CARD_3:def 4;

              then

              consider Px be set such that

               X50: x in Px & Px in ( rng ( ProjMap1 (LG,i))) by TARSKI:def 4;

              consider j be Element of NAT such that

               X51: Px = (( ProjMap1 (LG,i)) . j) by X50, FUNCT_2: 113;

              (( ProjMap1 (LG,i)) . j) = (LG . (i,j)) by MESFUNC9:def 6;

              then x in ((D . (i + 1)) /\ (G . j)) by X50, X51, D5;

              hence x in (D . (i + 1)) by XBOOLE_0:def 4;

            end;

            then ( Union ( ProjMap1 (LG,i))) c= (D . (i + 1)) by TARSKI:def 3;

            hence thesis by X48, XBOOLE_0:def 10;

          end;

          

           X5: for i be Element of NAT st i < ( len D) holds ((P * D) . (i + 1)) <= (( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) . i)

          proof

            let i be Element of NAT ;

            assume

             X50: i < ( len D);

            then

             X50a: 1 <= (i + 1) & (i + 1) <= ( len D) by NAT_1: 11, NAT_1: 13;

            then

             X51: (i + 1) in ( dom D) by FINSEQ_3: 25;

            then

             X52: (D . (i + 1)) in ( rng D) by FUNCT_1: 3;

            now

              let x,y be object;

              assume

               V1: x <> y;

              per cases ;

                suppose not x in ( dom ( ProjMap1 (LG,i))) or not y in ( dom ( ProjMap1 (LG,i)));

                then (( ProjMap1 (LG,i)) . x) = {} or (( ProjMap1 (LG,i)) . y) = {} by FUNCT_1:def 2;

                hence (( ProjMap1 (LG,i)) . x) misses (( ProjMap1 (LG,i)) . y) by XBOOLE_1: 65;

              end;

                suppose x in ( dom ( ProjMap1 (LG,i))) & y in ( dom ( ProjMap1 (LG,i)));

                then

                reconsider x1 = x, y1 = y as Element of NAT ;

                (( ProjMap1 (LG,i)) . x) = (LG . (i,x1)) & (( ProjMap1 (LG,i)) . y) = (LG . (i,y1)) by MESFUNC9:def 6;

                then (( ProjMap1 (LG,i)) . x) = ((D . (i + 1)) /\ (G . x1)) & (( ProjMap1 (LG,i)) . y) = ((D . (i + 1)) /\ (G . y1)) by X50a, D5;

                hence (( ProjMap1 (LG,i)) . x) misses (( ProjMap1 (LG,i)) . y) by V1, PROB_2:def 2, XBOOLE_1: 76;

              end;

            end;

            then

             X53: ( ProjMap1 (LG,i)) is disjoint_valued Function of NAT , S by PROB_2:def 2;

            

             X54: (D . (i + 1)) = ( Union ( ProjMap1 (LG,i))) by X4, X50;

            

             X55: ((P * D) . (i + 1)) = (P . (D . (i + 1))) by X51, FUNCT_1: 13

            .= (P . ( Union ( ProjMap1 (LG,i)))) by X4, X50;

             X56:

            now

              let k be Element of NAT ;

              (P . (LG . (i,k))) = ((P * LG) . (i,k)) by ZFMISC_1: 87, FUNCT_2: 15;

              

              then (( ProjMap1 ((P * LG),i)) . k) = (P . (LG . (i,k))) by MESFUNC9:def 6

              .= (P . (( ProjMap1 (LG,i)) . k)) by MESFUNC9:def 6;

              hence ((P * ( ProjMap1 (LG,i))) . k) = (( ProjMap1 ((P * LG),i)) . k) by FUNCT_2: 15;

            end;

            ( SUM (P * ( ProjMap1 (LG,i)))) = ( Sum (P * ( ProjMap1 (LG,i)))) by MEASURE8: 2

            .= ( lim ( Partial_Sums (P * ( ProjMap1 (LG,i))))) by MESFUNC9:def 3

            .= ( lim ( Partial_Sums ( ProjMap1 ((P * LG),i)))) by X56, FUNCT_2:def 8

            .= ( lim ( ProjMap1 (( Partial_Sums_in_cod2 (P * LG)),i))) by DBLSEQ_3: 53

            .= (( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) . i) by DBLSEQ_3:def 13;

            hence thesis by X55, X52, X53, X54, Def8;

          end;

          defpred SPD[ Nat] means $1 < ( len (P * D)) implies (SumPD . ($1 + 1)) <= (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . $1);

          ( rng D) c= S;

          then ( rng D) c= ( dom P) by FUNCT_2:def 1;

          then ( dom (P * D)) = ( dom D) by RELAT_1: 27;

          then

           X71: ( len (P * D)) = ( len D) by FINSEQ_3: 29;

          now

            assume

             X60: 0 < ( len (P * D));

            then

             X61: ((P * D) . ( 0 + 1)) <= (( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) . 0 ) by X5, X71;

            (SumPD . ( 0 + 1)) = ((SumPD . 0 ) + ((P * D) . ( 0 + 1))) by X60, X3;

            then (SumPD . ( 0 + 1)) = ((P * D) . 1) by X3, XXREAL_3: 4;

            hence (SumPD . ( 0 + 1)) <= (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . 0 ) by X61, MESFUNC9:def 1;

          end;

          then

           X62: SPD[ 0 ];

          

           X63: for k be Nat st SPD[k] holds SPD[(k + 1)]

          proof

            let k be Nat;

            assume

             X64: SPD[k];

            assume

             X65: (k + 1) < ( len (P * D));

            then

             X67: ((P * D) . ((k + 1) + 1)) <= (( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) . (k + 1)) by X5, X71;

            (SumPD . ((k + 1) + 1)) = ((SumPD . (k + 1)) + ((P * D) . ((k + 1) + 1))) by X3, X65;

            then (SumPD . ((k + 1) + 1)) <= ((( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . k) + (( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) . (k + 1))) by NAT_1: 13, X67, X64, X65, XXREAL_3: 36;

            hence (SumPD . ((k + 1) + 1)) <= (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . (k + 1)) by MESFUNC9:def 1;

          end;

          

           X68: for k be Nat holds SPD[k] from NAT_1:sch 2( X62, X63);

           XX70:

          now

            assume D = {} ;

            then ( union ( rng D)) = {} by ZFMISC_1: 2;

            then

             X69: ( union ( rng FSets)) = {} by B1, B2, CARD_3:def 4;

            ( union {a}) c= ( union ( rng FSets)) by E6, ZFMISC_1: 31, ZFMISC_1: 77;

            hence contradiction by X69;

          end;

          then

          consider i1 be Nat such that

           X70: ( len D) = (i1 + 1) by NAT_1: 6;

          reconsider i1 as Element of NAT by ORDINAL1:def 12;

          i1 < ( len D) by X70, NAT_1: 13;

          then

           X72: ( Sum (P * D)) <= (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . i1) by X70, X71, X68, X3;

          

           X73: ( len (P * D)) >= i1 by X70, X71, NAT_1: 11;

          

           W3: ( Partial_Sums_in_cod2 (P * LG)) is convergent_in_cod2 by DBLSEQ_3: 66;

          then

           X80: ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) is nonnegative by DBLSEQ_3: 65;

          then

           X74: (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . i1) <= (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . ( len (P * D))) by X73, RINFSUP2: 7, MESFUNC9: 16;

          (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . ( len (P * D))) = (( lim_in_cod2 ( Partial_Sums (P * LG))) . ( len (P * D)))

          proof

            per cases ;

              suppose

               X75: (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . ( len (P * D))) = +infty ;

              then ex k be Element of NAT st k <= ( len (P * D)) & ( ProjMap1 (( Partial_Sums_in_cod2 (P * LG)),k)) is convergent_to_+infty by DBLSEQ_3: 74;

              then ( lim ( ProjMap1 (( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 (P * LG))),( len (P * D))))) = +infty by DBLSEQ_3: 77;

              then ( lim ( ProjMap1 (( Partial_Sums (P * LG)),( len (P * D))))) = +infty by DBLSEQ_3:def 16;

              hence (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . ( len (P * D))) = (( lim_in_cod2 ( Partial_Sums (P * LG))) . ( len (P * D))) by X75, DBLSEQ_3:def 13;

            end;

              suppose (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . ( len (P * D))) <> +infty ;

              then

               X81: (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . ( len (P * D))) < +infty by XXREAL_0: 4;

              for m be Element of NAT holds ( ProjMap1 (( Partial_Sums_in_cod2 (P * LG)),m)) is convergent_to_finite_number

              proof

                let m be Element of NAT ;

                

                 W5: ( ProjMap1 (( Partial_Sums_in_cod2 (P * LG)),m)) is convergent_to_+infty or ( ProjMap1 (( Partial_Sums_in_cod2 (P * LG)),m)) is convergent_to_finite_number or ( ProjMap1 (( Partial_Sums_in_cod2 (P * LG)),m)) is convergent_to_-infty by W3, DBLSEQ_3:def 11, MESFUNC5:def 11;

                per cases ;

                  suppose m <= ( len (P * D));

                  then

                   W1: (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . m) <= (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . ( len (P * D))) by X80, MESFUNC9: 16, RINFSUP2: 7;

                  (( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) . m) <= (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . m) by X80, DBLSEQ_3: 4;

                  then

                   W2: (( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) . m) <= (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . ( len (P * D))) by W1, XXREAL_0: 2;

                  (( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) . m) >= 0 by X80, SUPINF_2: 51;

                  then (( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) . m) in REAL by W2, X81, XXREAL_0: 14;

                  then ( lim ( ProjMap1 (( Partial_Sums_in_cod2 (P * LG)),m))) in REAL by DBLSEQ_3:def 13;

                  hence ( ProjMap1 (( Partial_Sums_in_cod2 (P * LG)),m)) is convergent_to_finite_number by W5, MESFUNC5:def 12;

                end;

                  suppose m > ( len (P * D));

                  then

                  consider j be Nat such that

                   W7: m = (( len (P * D)) + j) by NAT_1: 10;

                  defpred H[ Nat] means (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . (( len (P * D)) + $1)) = (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . ( len (P * D)));

                  

                   W8: H[ 0 ];

                  

                   W9: for i be Nat st H[i] holds H[(i + 1)]

                  proof

                    let i be Nat;

                    assume

                     W12: H[i];

                    

                     W13: (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . ((( len (P * D)) + i) + 1)) = ((( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . ( len (P * D))) + (( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) . ((( len (P * D)) + i) + 1))) by W12, MESFUNC9:def 1;

                    for s be Nat holds (( ProjMap1 (( Partial_Sums_in_cod2 (P * LG)),((( len (P * D)) + i) + 1))) . s) = 0

                    proof

                      let s be Nat;

                      reconsider s1 = s as Element of NAT by ORDINAL1:def 12;

                      

                       W15: (( ProjMap1 (( Partial_Sums_in_cod2 (P * LG)),((( len (P * D)) + i) + 1))) . s1) = (( Partial_Sums_in_cod2 (P * LG)) . (((( len (P * D)) + i) + 1),s)) by MESFUNC9:def 6;

                      

                       P0: for k,j be Nat holds ((P * LG) . (((( len (P * D)) + k) + 1),j)) = 0

                      proof

                        let k,j be Nat;

                        reconsider k1 = k, j1 = j as Element of NAT by ORDINAL1:def 12;

                        ((( len D) + k) + 1) >= ( len D) by NAT_1: 11, NAT_1: 12;

                        then

                         P1: (((( len D) + k1) + 1) + 1) > ( len D) by NAT_1: 13;

                         [((( len (P * D)) + k1) + 1), j1] in [: NAT , NAT :] by ZFMISC_1: 87;

                        then [((( len (P * D)) + k1) + 1), j1] in ( dom LG) by FUNCT_2:def 1;

                        then ((P * LG) . (((( len (P * D)) + k) + 1),j)) = (P . (LG . (((( len D) + k1) + 1),j1))) by X71, FUNCT_1: 13;

                        then ((P * LG) . (((( len (P * D)) + k) + 1),j)) = (P . {} ) by D5, P1;

                        hence thesis by VALUED_0:def 19;

                      end;

                      defpred G[ Nat] means (( Partial_Sums_in_cod2 (P * LG)) . (((( len (P * D)) + i) + 1),$1)) = 0 ;

                      (( Partial_Sums_in_cod2 (P * LG)) . (((( len (P * D)) + i) + 1), 0 )) = ((P * LG) . (((( len (P * D)) + i) + 1), 0 )) by DBLSEQ_3:def 14;

                      then

                       W16: G[ 0 ] by P0;

                      

                       W17: for j be Nat st G[j] holds G[(j + 1)]

                      proof

                        let j be Nat;

                        assume

                         P2: G[j];

                        (( Partial_Sums_in_cod2 (P * LG)) . (((( len (P * D)) + i) + 1),(j + 1))) = ((( Partial_Sums_in_cod2 (P * LG)) . (((( len (P * D)) + i) + 1),j)) + ((P * LG) . (((( len (P * D)) + i) + 1),(j + 1)))) by DBLSEQ_3:def 14

                        .= ((P * LG) . (((( len (P * D)) + i) + 1),(j + 1))) by P2, XXREAL_3: 4;

                        hence G[(j + 1)] by P0;

                      end;

                      for j be Nat holds G[j] from NAT_1:sch 2( W16, W17);

                      hence (( ProjMap1 (( Partial_Sums_in_cod2 (P * LG)),((( len (P * D)) + i) + 1))) . s) = 0 by W15;

                    end;

                    then ( lim ( ProjMap1 (( Partial_Sums_in_cod2 (P * LG)),((( len (P * D)) + i) + 1)))) = 0 by MESFUNC5: 52;

                    then (( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) . ((( len (P * D)) + i) + 1)) = 0 by DBLSEQ_3:def 13;

                    hence H[(i + 1)] by W13, XXREAL_3: 4;

                  end;

                  for i be Nat holds H[i] from NAT_1:sch 2( W8, W9);

                  then H[j];

                  then

                   W10: (( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) . m) <= (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . ( len (P * D))) by W7, X80, DBLSEQ_3: 4;

                  (( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) . m) > -infty by X80, SUPINF_2: 51;

                  then (( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG))) . m) in REAL by W10, X81, XXREAL_0: 14;

                  then ( lim ( ProjMap1 (( Partial_Sums_in_cod2 (P * LG)),m))) in REAL by DBLSEQ_3:def 13;

                  hence ( ProjMap1 (( Partial_Sums_in_cod2 (P * LG)),m)) is convergent_to_finite_number by W5, MESFUNC5:def 12;

                end;

              end;

              then ( Partial_Sums (P * LG)) is convergent_in_cod2_to_finite by DBLSEQ_3:def 10, DBLSEQ_3: 81;

              

              then (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . ( len (P * D))) = ( lim ( ProjMap1 (( Partial_Sums_in_cod2 ( Partial_Sums_in_cod1 (P * LG))),( len (P * D))))) by DBLSEQ_3: 82

              .= ( lim ( ProjMap1 (( Partial_Sums (P * LG)),( len (P * D))))) by DBLSEQ_3:def 16;

              hence (( Partial_Sums ( lim_in_cod2 ( Partial_Sums_in_cod2 (P * LG)))) . ( len (P * D))) = (( lim_in_cod2 ( Partial_Sums (P * LG))) . ( len (P * D))) by DBLSEQ_3:def 13;

            end;

          end;

          then

           X100: ( Sum (P * D)) <= (( lim_in_cod2 ( Partial_Sums (P * LG))) . ( len (P * D))) by X74, X72, XXREAL_0: 2;

          for j be Nat holds (( Partial_Sums_in_cod1 (P * LG)) . (( len (P * D)),j)) = ((P * G) . j)

          proof

            let j be Nat;

            reconsider j1 = j as Element of NAT by ORDINAL1:def 12;

            consider k,m be Nat such that

             M0: m in ( dom (s . k)) & ((((( Partial_Sums ( Length s)) . k) - ( len (s . k))) + m) - 1) = j & (G . j) = ((s . k) . m) by Def4;

            reconsider k1 = k as Element of NAT by ORDINAL1:def 12;

             [k1, (s . k1)] in Z by F9;

            then

            consider k2 be Nat, Sk be disjoint_valued FinSequence of S such that

             M1: [k1, (s . k1)] = [k2, Sk] & ( Union Sk) = (FSets . k2) & ((FSets . k2) = {} implies Sk = <* {} *>);

            

             M2: (s . k) = Sk & ( Union Sk) = (FSets . k) & ((FSets . k) = {} implies Sk = <* {} *>) by M1, XTUPLE_0: 1;

            defpred Cj[ Nat, object] means $2 = ((D . $1) /\ (G . j));

            

             M3: for k be Nat st k in ( Seg ( len D)) holds ex x be Element of S st Cj[k, x]

            proof

              let k be Nat;

              assume

               M5: k in ( Seg ( len D));

              then 1 <= k & k <= ( len D) by FINSEQ_1: 1;

              then

              consider k1 be Nat such that

               M4: k = (k1 + 1) by NAT_1: 6;

              reconsider kk1 = k1 as Element of NAT by ORDINAL1:def 12;

              (LG . (kk1,j1)) = ((D . k) /\ (G . j)) by M4, M5, D5, FINSEQ_1: 1;

              hence thesis;

            end;

            consider Cj be FinSequence of S such that

             M7: ( dom Cj) = ( Seg ( len D)) & for k be Nat st k in ( Seg ( len D)) holds Cj[k, (Cj . k)] from FINSEQ_1:sch 5( M3);

            

             M7a: ( len Cj) = ( len D) by M7, FINSEQ_1:def 3;

            now

              let x,y be object;

              assume

               M61: x <> y;

              per cases ;

                suppose

                 M62: x in ( dom Cj) & y in ( dom Cj);

                then

                reconsider x1 = x, y1 = y as Nat;

                (Cj . x) = ((D . x1) /\ (G . j)) & (Cj . y) = ((D . y1) /\ (G . j)) by M7, M62;

                hence (Cj . x) misses (Cj . y) by M61, PROB_2:def 2, XBOOLE_1: 76;

              end;

                suppose not x in ( dom Cj) or not y in ( dom Cj);

                then (Cj . x) = {} or (Cj . y) = {} by FUNCT_1:def 2;

                hence (Cj . x) misses (Cj . y) by XBOOLE_1: 65;

              end;

            end;

            then

            reconsider Cj as disjoint_valued FinSequence of S by PROB_2:def 2;

            now

              let x be object;

              assume x in ( Union Cj);

              then x in ( union ( rng Cj)) by CARD_3:def 4;

              then

              consider V be set such that

               M64: x in V & V in ( rng Cj) by TARSKI:def 4;

              consider y be object such that

               M65: y in ( dom Cj) & V = (Cj . y) by M64, FUNCT_1:def 3;

              reconsider y as Nat by M65;

              (Cj . y) = ((D . y) /\ (G . j)) by M65, M7;

              hence x in (G . j) by M64, M65, XBOOLE_0:def 4;

            end;

            then

             M66: ( Union Cj) c= (G . j) by TARSKI:def 3;

            now

              let x be object;

              assume

               M67a: x in (G . j);

              then x in (Sk . m) & (Sk . m) in ( rng Sk) by M0, M2, FUNCT_1: 3;

              then x in ( union ( rng Sk)) by TARSKI:def 4;

              then

               M67: x in (FSets . k2) by M1, CARD_3:def 4;

              ( dom FSets) = NAT by FUNCT_2:def 1;

              then (FSets . k2) in ( rng FSets) by FUNCT_1: 3, ORDINAL1:def 12;

              then x in ( union ( rng FSets)) by M67, TARSKI:def 4;

              then x in ( union ( rng D)) by B1, B2, CARD_3:def 4;

              then

              consider V be set such that

               M68: x in V & V in ( rng D) by TARSKI:def 4;

              consider y be object such that

               M69: y in ( dom D) & V = (D . y) by M68, FUNCT_1:def 3;

              reconsider y as Nat by M69;

              

               M70: x in ((D . y) /\ (G . j)) by M67a, M68, M69, XBOOLE_0:def 4;

              y in ( Seg ( len D)) by M69, FINSEQ_1:def 3;

              then

               M71: x in (Cj . y) by M7, M70;

              y in ( dom Cj) by M69, FINSEQ_1:def 3, M7;

              then (Cj . y) in ( rng Cj) by FUNCT_1: 3;

              then x in ( union ( rng Cj)) by M71, TARSKI:def 4;

              hence x in ( Union Cj) by CARD_3:def 4;

            end;

            then (G . j) c= ( Union Cj) by TARSKI:def 3;

            then

             M6: ( Union Cj) = (G . j) by M66, XBOOLE_0:def 10;

            

             M6b: (P . (G . j)) = ( Sum (P * Cj)) by M6, Def8;

            j in NAT by ORDINAL1:def 12;

            then j in ( dom G) by FUNCT_2:def 1;

            then

             M6a: ((P * G) . j) = ( Sum (P * Cj)) by M6b, FUNCT_1: 13;

            consider SumPCj be sequence of ExtREAL such that

             M8: ( Sum (P * Cj)) = (SumPCj . ( len (P * Cj))) & (SumPCj . 0 ) = 0. & for i be Nat st i < ( len (P * Cj)) holds (SumPCj . (i + 1)) = ((SumPCj . i) + ((P * Cj) . (i + 1))) by EXTREAL1:def 2;

            ( rng Cj) c= S;

            then ( rng Cj) c= ( dom P) by FUNCT_2:def 1;

            then

             N9: ( dom (P * Cj)) = ( dom Cj) by RELAT_1: 27;

            then

             M9: ( len (P * Cj)) = ( len Cj) by FINSEQ_3: 29;

            

             M13: for i be Nat st i < ( len D) holds ((P * Cj) . (i + 1)) = ((P * LG) . (i,j))

            proof

              let i be Nat;

              assume i < ( len D);

              then

               M11: 1 <= (i + 1) & (i + 1) <= ( len D) by NAT_1: 11, NAT_1: 13;

              then

               M12: (i + 1) in ( Seg ( len D));

              ((P * Cj) . (i + 1)) = (P . (Cj . (i + 1))) by M11, N9, M7a, FINSEQ_3: 25, FUNCT_1: 12;

              then ((P * Cj) . (i + 1)) = (P . ((D . (i + 1)) /\ (G . j))) by M7, M12;

              then

               M10: ((P * Cj) . (i + 1)) = (P . (LG . (i,j))) by M11, D5a;

              i in NAT & j in NAT by ORDINAL1:def 12;

              then [i, j] in [: NAT , NAT :] by ZFMISC_1: 87;

              then [i, j] in ( dom LG) by FUNCT_2:def 1;

              hence ((P * Cj) . (i + 1)) = ((P * LG) . (i,j)) by M10, FUNCT_1: 13;

            end;

            

             MM15: (( len (P * D)) + 1) > ( len D) by X71, NAT_1: 13;

            ( len (P * D)) in NAT & j in NAT by ORDINAL1:def 12;

            then [( len (P * D)), j] in [: NAT , NAT :] by ZFMISC_1: 87;

            then [( len (P * D)), j] in ( dom LG) by FUNCT_2:def 1;

            then ((P * LG) . (( len (P * D)),j)) = (P . (LG . (( len (P * D)),j))) by FUNCT_1: 13;

            then ((P * LG) . (( len (P * D)),j)) = (P . {} ) by MM15, D5a;

            then

             M15: ((P * LG) . (( len (P * D)),j)) = 0 by VALUED_0:def 19;

            consider LENDM1 be Nat such that

             M23: ( len D) = (LENDM1 + 1) by XX70, NAT_1: 6;

            

             M24: LENDM1 < ( len (P * D)) by M23, X71, NAT_1: 13;

            defpred EQ[ Nat] means $1 < ( len (P * D)) implies (( Partial_Sums_in_cod1 (P * LG)) . ($1,j)) = (SumPCj . ($1 + 1));

            (( Partial_Sums_in_cod1 (P * LG)) . ( 0 ,j)) = ((P * LG) . ( 0 ,j)) by DBLSEQ_3:def 15;

            then

             M17: (( Partial_Sums_in_cod1 (P * LG)) . ( 0 ,j)) = ((P * Cj) . ( 0 + 1)) by XX70, M13;

            (SumPCj . ( 0 + 1)) = ( 0 + ((P * Cj) . ( 0 + 1))) by XX70, M7a, M9, M8;

            then

             M18: EQ[ 0 ] by M17, XXREAL_3: 4;

            

             M22: for k be Nat st EQ[k] holds EQ[(k + 1)]

            proof

              let k be Nat;

              assume

               M19: EQ[k];

              assume

               M20: (k + 1) < ( len (P * D));

              

              then (( Partial_Sums_in_cod1 (P * LG)) . ((k + 1),j)) = ((SumPCj . (k + 1)) + ((P * LG) . ((k + 1),j))) by M19, NAT_1: 13, DBLSEQ_3:def 15

              .= ((SumPCj . (k + 1)) + ((P * Cj) . ((k + 1) + 1))) by M20, M13, X71;

              hence (( Partial_Sums_in_cod1 (P * LG)) . ((k + 1),j)) = (SumPCj . ((k + 1) + 1)) by M20, M9, M7a, M8, X71;

            end;

            for k be Nat holds EQ[k] from NAT_1:sch 2( M18, M22);

            then (( Partial_Sums_in_cod1 (P * LG)) . (LENDM1,j)) = ((P * G) . j) by M6a, M8, M24, M23, M7a, M9;

            then (( Partial_Sums_in_cod1 (P * LG)) . (( len (P * D)),j)) = (((P * G) . j) + 0 ) by M15, X71, M23, DBLSEQ_3:def 15;

            hence thesis by XXREAL_3: 4;

          end;

          then

           X120: (M . ( Union FSets)) <= ( Sum (P * G)) by X2, X100, DBLSEQ_3: 41;

          ( Partial_Sums (P * G)) is non-decreasing by MESFUNC9: 16;

          then

           X123: ( Partial_Sums (P * G)) is convergent by RINFSUP2: 37;

          

           X124: ( Partial_Sums (M * FSets)) is subsequence of ( Partial_Sums (P * G))

          proof

            consider N be increasing sequence of NAT such that

             Z0: for k be Nat holds (N . k) = ((( Partial_Sums ( Length s)) . k) - 1) by Th11;

            defpred P[ Nat] means ((( Partial_Sums (P * G)) * N) . $1) = (( Partial_Sums (M * FSets)) . $1);

             [ 0 , (s . 0 )] in Z by F9;

            then

            consider n0 be Nat, E0 be disjoint_valued FinSequence of S such that

             Z1: [ 0 , (s . 0 )] = [n0, E0] & ( Union E0) = (FSets . n0) & ((FSets . n0) = {} implies E0 = <* {} *>);

            

             Z2: n0 = 0 & E0 = (s . 0 ) by Z1, XTUPLE_0: 1;

            

             Z4: (M . ( Union E0)) = ( Sum (P * E0)) by A1, Z1;

            consider SPE0 be sequence of ExtREAL such that

             Z5: ( Sum (P * E0)) = (SPE0 . ( len (P * E0))) & (SPE0 . 0 ) = 0. & for i be Nat st i < ( len (P * E0)) holds (SPE0 . (i + 1)) = ((SPE0 . i) + ((P * E0) . (i + 1))) by EXTREAL1:def 2;

            ( rng E0) c= S;

            then ( rng E0) c= ( dom P) by FUNCT_2:def 1;

            then

             ZZ10: ( dom (P * E0)) = ( dom E0) by RELAT_1: 27;

            then

             Z10: ( len (P * E0)) = ( len E0) by FINSEQ_3: 29;

            ( len (s . 0 )) >= 1 by FINSEQ_1: 20;

            then

             Z11: ( len (s . 0 )) in ( dom (s . 0 )) & 1 in ( dom (s . 0 )) by FINSEQ_3: 25;

            then

            consider N0 be Nat such that

             Z6: N0 = ((((( Partial_Sums ( Length s)) . 0 ) - ( len (s . 0 ))) + ( len (s . 0 ))) - 1) & (G . N0) = ((s . 0 ) . ( len (s . 0 ))) by Th13;

            

             Z6d: N0 = (N . 0 ) by Z0, Z6;

            N0 = ((( Length s) . 0 ) - 1) by Z6, SERIES_1:def 1;

            then

             Z6c: (N0 + 1) = ( len (s . 0 )) by Def3;

            then

             Z6b: N0 < ( len (s . 0 )) by NAT_1: 13;

            defpred P0[ Nat] means $1 < ( len (P * E0)) implies (( Partial_Sums (P * G)) . $1) = (SPE0 . ($1 + 1));

            consider z0 be Nat such that

             Z7: z0 = ((((( Partial_Sums ( Length s)) . 0 ) - ( len (s . 0 ))) + 1) - 1) & (G . z0) = ((s . 0 ) . 1) by Z11, Th13;

            z0 = ((( Length s) . 0 ) - ( len (s . 0 ))) by Z7, SERIES_1:def 1;

            then

             Z8: z0 = (( len (s . 0 )) - ( len (s . 0 ))) by Def3;

            

             Z12: (( Partial_Sums (P * G)) . 0 ) = ((P * G) . 0 ) by MESFUNC9:def 1

            .= (P . ((s . 0 ) . 1)) by Z7, Z8, FUNCT_2: 15

            .= ((P * E0) . 1) by Z11, Z2, FUNCT_1: 13;

            (SPE0 . ( 0 + 1)) = ( 0 + ((P * E0) . 1)) by Z5, Z10, Z2;

            then

             ZZ1: P0[ 0 ] by Z12, XXREAL_3: 4;

            

             ZZ2: for i be Nat st P0[i] holds P0[(i + 1)]

            proof

              let i be Nat;

              assume

               Z13: P0[i];

              assume

               Z14: (i + 1) < ( len (P * E0));

              

               Z16: (( Partial_Sums (P * G)) . (i + 1)) = ((( Partial_Sums (P * G)) . i) + ((P * G) . (i + 1))) by MESFUNC9:def 1

              .= ((SPE0 . (i + 1)) + (P . (G . (i + 1)))) by Z14, NAT_1: 13, Z13, FUNCT_2: 15;

              

               Z18: 1 <= ((i + 1) + 1) & ((i + 1) + 1) <= ( len (P * E0)) by Z14, NAT_1: 11, NAT_1: 13;

              then

              consider zi1 be Nat such that

               Z17: zi1 = ((((( Partial_Sums ( Length s)) . 0 ) - ( len (s . 0 ))) + ((i + 1) + 1)) - 1) & (G . zi1) = ((s . 0 ) . ((i + 1) + 1)) by Th13, ZZ10, Z2, FINSEQ_3: 25;

              zi1 = ((((( Length s) . 0 ) - ( len (s . 0 ))) + ((i + 1) + 1)) - 1) by Z17, SERIES_1:def 1

              .= (((( len (s . 0 )) - ( len (s . 0 ))) + ((i + 1) + 1)) - 1) by Def3;

              then (P . (G . (i + 1))) = ((P * E0) . ((i + 1) + 1)) by Z17, Z18, ZZ10, Z2, FINSEQ_3: 25, FUNCT_1: 13;

              hence (( Partial_Sums (P * G)) . (i + 1)) = (SPE0 . ((i + 1) + 1)) by Z14, Z5, Z16;

            end;

            for i be Nat holds P0[i] from NAT_1:sch 2( ZZ1, ZZ2);

            then (( Partial_Sums (P * G)) . N0) = (M . (FSets . 0 )) by Z6b, Z6c, Z2, Z10, Z5, Z4, Z1;

            then (( Partial_Sums (P * G)) . N0) = ((M * FSets) . 0 ) by FUNCT_2: 15;

            then (( Partial_Sums (P * G)) . N0) = (( Partial_Sums (M * FSets)) . 0 ) by MESFUNC9:def 1;

            then

             Z100: P[ 0 ] by Z6d, FUNCT_2: 15;

            

             Z101: for n be Nat st P[n] holds P[(n + 1)]

            proof

              let n be Nat;

              assume

               Z20: P[n];

               [(n + 1), (s . (n + 1))] in Z by F9;

              then

              consider N1 be Nat, E be disjoint_valued FinSequence of S such that

               Z21: [(n + 1), (s . (n + 1))] = [N1, E] & ( Union E) = (FSets . N1) & ((FSets . N1) = {} implies E = <* {} *>);

              

               Z22: (n + 1) = N1 & (s . (n + 1)) = E by Z21, XTUPLE_0: 1;

              

               Z24: (M . ( Union E)) = ( Sum (P * E)) by A1, Z21;

              consider SPE be sequence of ExtREAL such that

               Z25: ( Sum (P * E)) = (SPE . ( len (P * E))) & (SPE . 0 ) = 0. & for i be Nat st i < ( len (P * E)) holds (SPE . (i + 1)) = ((SPE . i) + ((P * E) . (i + 1))) by EXTREAL1:def 2;

              ( rng E) c= S;

              then ( rng E) c= ( dom P) by FUNCT_2:def 1;

              then

               ZZ30: ( dom (P * E)) = ( dom E) by RELAT_1: 27;

              then

               Z30: ( len (P * E)) = ( len E) by FINSEQ_3: 29;

              ( len (s . (n + 1))) >= 1 by FINSEQ_1: 20;

              then

               Z31: ( len (s . (n + 1))) in ( dom (s . (n + 1))) & 1 in ( dom (s . (n + 1))) by FINSEQ_3: 25;

              then

              consider NEnd be Nat such that

               Z26: NEnd = ((((( Partial_Sums ( Length s)) . (n + 1)) - ( len (s . (n + 1)))) + ( len (s . (n + 1)))) - 1) & (G . NEnd) = ((s . (n + 1)) . ( len (s . (n + 1)))) by Th13;

              

               Z26d: NEnd = (N . (n + 1)) by Z0, Z26;

              consider NSt be Nat such that

               Z27: NSt = ((((( Partial_Sums ( Length s)) . (n + 1)) - ( len (s . (n + 1)))) + 1) - 1) & (G . NSt) = ((s . (n + 1)) . 1) by Z31, Th13;

              NSt = (((( Partial_Sums ( Length s)) . n) + (( Length s) . (n + 1))) - ( len (s . (n + 1)))) by Z27, SERIES_1:def 1;

              then

               Z28: NSt = (((( Partial_Sums ( Length s)) . n) + ( len (s . (n + 1)))) - ( len (s . (n + 1)))) by Def3;

              

               Z50: (N . n) = ((( Partial_Sums ( Length s)) . n) - 1) by Z0;

              defpred PE[ Nat] means $1 < ( len (P * E)) implies (( Partial_Sums (P * G)) . (((N . n) + $1) + 1)) = ((( Partial_Sums (P * G)) . (N . n)) + (SPE . ($1 + 1)));

              

               Z40: (( Partial_Sums (P * G)) . ((N . n) + 1)) = ((( Partial_Sums (P * G)) . (N . n)) + ((P * G) . ((N . n) + 1))) by MESFUNC9:def 1

              .= ((( Partial_Sums (P * G)) . (N . n)) + (P . (G . ((N . n) + 1)))) by FUNCT_2: 15

              .= ((( Partial_Sums (P * G)) . (N . n)) + ((P * E) . 1)) by Z31, Z22, Z50, Z28, Z27, FUNCT_1: 13;

              (SPE . ( 0 + 1)) = ((SPE . 0 ) + ((P * E) . ( 0 + 1))) by Z25, Z30, Z22;

              then

               Z60: PE[ 0 ] by Z40, Z25, XXREAL_3: 4;

              

               Z61: for j be Nat st PE[j] holds PE[(j + 1)]

              proof

                let j be Nat;

                assume

                 Z52: PE[j];

                assume

                 Z53: (j + 1) < ( len (P * E));

                then

                 Z58a: 1 <= ((j + 1) + 1) & ((j + 1) + 1) <= ( len (P * E)) by NAT_1: 11, NAT_1: 13;

                then

                consider Nj be Nat such that

                 Z58: Nj = ((((( Partial_Sums ( Length s)) . (n + 1)) - ( len (s . (n + 1)))) + ((j + 1) + 1)) - 1) & (G . Nj) = ((s . (n + 1)) . ((j + 1) + 1)) by Th13, ZZ30, Z22, FINSEQ_3: 25;

                

                 Z55: (( Partial_Sums (P * G)) . (N . n)) > -infty by SUPINF_2: 51;

                

                 Z56: (P . (G . ((((N . n) + j) + 1) + 1))) > -infty by SUPINF_2: 51;

                defpred SP[ Nat] means $1 <= ( len (P * E)) implies (SPE . $1) >= 0 ;

                

                 ZZ1: SP[ 0 ] by Z25;

                

                 ZZ2: for t be Nat st SP[t] holds SP[(t + 1)]

                proof

                  let t be Nat;

                  assume

                   ZZ3: SP[t];

                  assume

                   ZZ6: (t + 1) <= ( len (P * E));

                  then

                   ZZ5: (SPE . (t + 1)) = ((SPE . t) + ((P * E) . (t + 1))) by Z25, NAT_1: 13;

                  (t + 1) in ( dom (P * E)) by NAT_1: 11, ZZ6, FINSEQ_3: 25;

                  then ((P * E) . (t + 1)) = (P . (E . (t + 1))) by FUNCT_1: 12;

                  then ((P * E) . (t + 1)) >= 0 by SUPINF_2: 51;

                  hence thesis by ZZ3, ZZ6, NAT_1: 13, ZZ5;

                end;

                for t be Nat holds SP[t] from NAT_1:sch 2( ZZ1, ZZ2);

                then

                 Z57: (SPE . (j + 1)) >= 0 by Z53;

                (( Partial_Sums (P * G)) . (((N . n) + (j + 1)) + 1)) = ((( Partial_Sums (P * G)) . ((N . n) + (j + 1))) + ((P * G) . (((N . n) + (j + 1)) + 1))) by MESFUNC9:def 1

                .= (((( Partial_Sums (P * G)) . (N . n)) + (SPE . (j + 1))) + (P . (G . ((((N . n) + j) + 1) + 1)))) by Z53, Z52, NAT_1: 13, FUNCT_2: 15

                .= ((( Partial_Sums (P * G)) . (N . n)) + ((SPE . (j + 1)) + (P . (G . ((((N . n) + j) + 1) + 1))))) by Z55, Z56, Z57, XXREAL_3: 29

                .= ((( Partial_Sums (P * G)) . (N . n)) + ((SPE . (j + 1)) + ((P * E) . ((j + 1) + 1)))) by Z58a, Z58, Z50, Z28, Z27, Z22, ZZ30, FINSEQ_3: 25, FUNCT_1: 13;

                hence thesis by Z53, Z25;

              end;

              

               Z62: for j be Nat holds PE[j] from NAT_1:sch 2( Z60, Z61);

              

               Z59a: ((N . n) + ( len (P * E))) = (((( Partial_Sums ( Length s)) . n) - 1) + ( len (s . (n + 1)))) by Z0, Z30, Z22

              .= (((( Partial_Sums ( Length s)) . n) - 1) + (( Length s) . (n + 1))) by Def3

              .= (((( Partial_Sums ( Length s)) . n) + (( Length s) . (n + 1))) - 1)

              .= (N . (n + 1)) by Z26, Z26d, SERIES_1:def 1;

              consider sn1 be Nat such that

               Z63: ( len (P * E)) = (sn1 + 1) by Z22, Z30, NAT_1: 6;

              sn1 < ( len (P * E)) by Z63, NAT_1: 13;

              

              then

               TA: (( Partial_Sums (P * G)) . (((N . n) + sn1) + 1)) = ((( Partial_Sums (P * G)) . (N . n)) + ( Sum (P * E))) by Z25, Z62, Z63

              .= ((( Partial_Sums (P * G)) . (N . n)) + ((M * FSets) . (n + 1))) by Z24, Z21, Z22, FUNCT_2: 15;

              (( Partial_Sums (M * FSets)) . (n + 1)) = (((( Partial_Sums (P * G)) * N) . n) + ((M * FSets) . (n + 1))) by Z20, MESFUNC9:def 1

              .= (( Partial_Sums (P * G)) . (N . (n + 1))) by TA, Z59a, Z63, ORDINAL1:def 12, FUNCT_2: 15

              .= ((( Partial_Sums (P * G)) * N) . (n + 1)) by FUNCT_2: 15;

              hence thesis;

            end;

            for n be Nat holds P[n] from NAT_1:sch 2( Z100, Z101);

            then for n be Element of NAT holds (( Partial_Sums (M * FSets)) . n) = ((( Partial_Sums (P * G)) * N) . n);

            hence ( Partial_Sums (M * FSets)) is subsequence of ( Partial_Sums (P * G)) by FUNCT_2:def 8;

          end;

          

           X125: ( Sum (M * FSets)) = ( Sum (P * G))

          proof

            per cases by X123, MESFUNC5:def 11, MESFUNC9: 8;

              suppose

               L1: ( Partial_Sums (P * G)) is convergent_to_+infty;

              then ( lim ( Partial_Sums (M * FSets))) = +infty by X124, DBLSEQ_3: 27;

              then ( lim ( Partial_Sums (M * FSets))) = ( lim ( Partial_Sums (P * G))) by L1, MESFUNC9: 7;

              then ( Sum (M * FSets)) = ( lim ( Partial_Sums (P * G))) by MESFUNC9:def 3;

              hence ( Sum (M * FSets)) = ( Sum (P * G)) by MESFUNC9:def 3;

            end;

              suppose ( Partial_Sums (P * G)) is convergent_to_finite_number;

              then ( lim ( Partial_Sums (M * FSets))) = ( lim ( Partial_Sums (P * G))) by X124, DBLSEQ_3: 26;

              then ( Sum (M * FSets)) = ( lim ( Partial_Sums (P * G))) by MESFUNC9:def 3;

              hence ( Sum (M * FSets)) = ( Sum (P * G)) by MESFUNC9:def 3;

            end;

          end;

          

           H0: ( Partial_Sums (M * FSets)) is non-decreasing by MESFUNC9: 16;

          for n be Nat holds (( Partial_Sums (M * FSets)) . n) <= (M . ( union ( rng FSets)))

          proof

            let n be Nat;

            

             H1: ( union ( rng (FSets | ( Segm (n + 1))))) in ( Field_generated_by S) by Th58;

            ( rng (FSets | ( Segm (n + 1)))) c= ( rng FSets) by RELAT_1: 70;

            then (M . ( union ( rng (FSets | ( Segm (n + 1)))))) <= (M . ( union ( rng FSets))) by B0, H1, MEASURE1: 8, ZFMISC_1: 77;

            hence (( Partial_Sums (M * FSets)) . n) <= (M . ( union ( rng FSets))) by Th58;

          end;

          then ( lim ( Partial_Sums (M * FSets))) <= (M . ( union ( rng FSets))) by H0, RINFSUP2: 37, MESFUNC9: 9;

          then ( Sum (M * FSets)) <= (M . ( union ( rng FSets))) by MESFUNC9:def 3;

          then ( Sum (M * FSets)) <= (M . ( Union FSets)) by CARD_3:def 4;

          then

           X126: (M . ( Union FSets)) = ( Sum (M * FSets)) by X125, X120, XXREAL_0: 1;

          ( Sum (M * FSets)) = ( SUM (M * FSets)) by MEASURE8: 2;

          hence ( SUM (M * FSets)) = (M . ( union ( rng FSets))) by X126, CARD_3:def 4;

        end;

          suppose

           LL1: ( rng FSets) is empty-membered;

          then ( union ( rng FSets)) = {} by Th52;

          then

           L2: (M . ( union ( rng FSets))) = 0 by VALUED_0:def 19;

          

           LL3: for n be Nat holds ((M * FSets) . n) = 0

          proof

            let n be Nat;

            

             LL4: ( dom FSets) = NAT by FUNCT_2:def 1;

            then (FSets . n) in ( rng FSets) by FUNCT_1: 3, ORDINAL1:def 12;

            then (FSets . n) = {} by LL1;

            then ((M * FSets) . n) = (M . {} ) by LL4, ORDINAL1:def 12, FUNCT_1: 13;

            hence ((M * FSets) . n) = 0 by VALUED_0:def 19;

          end;

          

           LL5: ( dom ( Partial_Sums (M * FSets))) = NAT & ( dom ( seq_const 0 )) = NAT by FUNCT_2:def 1;

          for n be object st n in ( dom ( Partial_Sums (M * FSets))) holds (( Partial_Sums (M * FSets)) . n) = (( seq_const 0 ) . n)

          proof

            let n be object;

            assume n in ( dom ( Partial_Sums (M * FSets)));

            then

            reconsider n1 = n as Nat;

            defpred P[ Nat] means (( Partial_Sums (M * FSets)) . $1) = 0 ;

            (( Partial_Sums (M * FSets)) . 0 ) = ((M * FSets) . 0 ) by MESFUNC9:def 1;

            then

             LL8: P[ 0 ] by LL3;

            

             LL9: for i be Nat st P[i] holds P[(i + 1)]

            proof

              let i be Nat;

              assume P[i];

              then ((( Partial_Sums (M * FSets)) . i) + ((M * FSets) . (i + 1))) = ((M * FSets) . (i + 1)) by XXREAL_3: 4;

              then (( Partial_Sums (M * FSets)) . (i + 1)) = ((M * FSets) . (i + 1)) by MESFUNC9:def 1;

              hence P[(i + 1)] by LL3;

            end;

            for i be Nat holds P[i] from NAT_1:sch 2( LL8, LL9);

            then (( Partial_Sums (M * FSets)) . n1) = 0 ;

            hence thesis;

          end;

          then ( Partial_Sums (M * FSets)) = ( seq_const 0 ) by LL5, FUNCT_1:def 11;

          then

           L4: ( Partial_Sums (M * FSets)) is convergent_to_finite_number & ( Partial_Sums (M * FSets)) is convergent & ( lim ( Partial_Sums (M * FSets))) = ( lim ( seq_const 0 )) by RINFSUP2: 14;

          ( SUM (M * FSets)) = ( Sum (M * FSets)) by MEASURE8: 2;

          hence ( SUM (M * FSets)) = (M . ( union ( rng FSets))) by L2, L4, MESFUNC9:def 3;

        end;

      end;

      hence M is completely-additive by MEASURE8:def 11;

    end;

    definition

      let X be set, S be semialgebra_of_sets of X, P be pre-Measure of S;

      :: MEASURE9:def8

      mode induced_Measure of S,P -> Measure of ( Field_generated_by S) means

      : Def9: for A be set st A in ( Field_generated_by S) holds for F be disjoint_valued FinSequence of S st A = ( Union F) holds (it . A) = ( Sum (P * F));

      existence by Th55;

    end

    theorem :: MEASURE9:60

    

     Th60: for X be set, S be semialgebra_of_sets of X, P be pre-Measure of S, M be induced_Measure of S, P holds M is completely-additive

    proof

      let X be set, S be semialgebra_of_sets of X, P be pre-Measure of S, M be induced_Measure of S, P;

      for A be set st A in ( Field_generated_by S) holds for F be disjoint_valued FinSequence of S st A = ( Union F) holds (M . A) = ( Sum (P * F)) by Def9;

      hence thesis by Th59;

    end;

    theorem :: MEASURE9:61

    

     Th61: for X be non empty set, S be semialgebra_of_sets of X, P be pre-Measure of S, M be induced_Measure of S, P holds (( sigma_Meas ( C_Meas M)) | ( sigma ( Field_generated_by S))) is sigma_Measure of ( sigma ( Field_generated_by S))

    proof

      let X be non empty set, S be semialgebra_of_sets of X, P be pre-Measure of S, M be induced_Measure of S, P;

      M is completely-additive by Th60;

      then

      consider N be sigma_Measure of ( sigma ( Field_generated_by S)) such that

       A1: N is_extension_of M & N = (( sigma_Meas ( C_Meas M)) | ( sigma ( Field_generated_by S))) by MEASURE8: 33;

      thus thesis by A1;

    end;

    definition

      let X be non empty set, S be semialgebra_of_sets of X, P be pre-Measure of S, M be induced_Measure of S, P;

      :: MEASURE9:def9

      mode induced_sigma_Measure of S,M -> sigma_Measure of ( sigma ( Field_generated_by S)) means

      : Def10: it = (( sigma_Meas ( C_Meas M)) | ( sigma ( Field_generated_by S)));

      existence

      proof

        (( sigma_Meas ( C_Meas M)) | ( sigma ( Field_generated_by S))) is sigma_Measure of ( sigma ( Field_generated_by S)) by Th61;

        hence thesis;

      end;

    end

    theorem :: MEASURE9:62

    for X be non empty set, S be semialgebra_of_sets of X, P be pre-Measure of S, m be induced_Measure of S, P, M be induced_sigma_Measure of S, m holds M is_extension_of m

    proof

      let X be non empty set, S be semialgebra_of_sets of X, P be pre-Measure of S, m be induced_Measure of S, P, M be induced_sigma_Measure of S, m;

      m is completely-additive by Th60;

      then

      consider N be sigma_Measure of ( sigma ( Field_generated_by S)) such that

       A2: N is_extension_of m & N = (( sigma_Meas ( C_Meas m)) | ( sigma ( Field_generated_by S))) by MEASURE8: 33;

      thus M is_extension_of m by A2, Def10;

    end;