jordan1g.miz



    begin

    reserve n for Nat;

    registration

      cluster trivial for FinSequence;

      existence

      proof

        take {} ;

        thus thesis;

      end;

    end

    theorem :: JORDAN1G:1

    

     Th1: for f be trivial FinSequence holds f is empty or ex x be object st f = <*x*>

    proof

      let f be trivial FinSequence;

      assume f is non empty;

      then

      consider x be object such that

       A1: f = {x} by ZFMISC_1: 131;

      x in {x} by TARSKI:def 1;

      then

      consider y,z be object such that

       A2: x = [y, z] by A1, RELAT_1:def 1;

      

       A3: 1 in ( dom f) by A1, FINSEQ_5: 6;

      take z;

      ( dom f) = {y} by A1, A2, RELAT_1: 9;

      then 1 = y by A3, TARSKI:def 1;

      hence thesis by A1, A2, FINSEQ_1:def 5;

    end;

    registration

      let p be non trivial FinSequence;

      cluster ( Rev p) -> non trivial;

      coherence

      proof

        assume

         A1: ( Rev p) is trivial;

        per cases by A1, Th1;

          suppose ( Rev p) is empty;

          hence contradiction;

        end;

          suppose ex x be object st ( Rev p) = <*x*>;

          then

          consider x be object such that

           A2: ( Rev p) = <*x*>;

          p = ( Rev <*x*>) by A2

          .= <*x*> by FINSEQ_5: 60;

          hence contradiction;

        end;

      end;

    end

    theorem :: JORDAN1G:2

    

     Th2: for D be non empty set holds for f be FinSequence of D holds for G be Matrix of D holds for p be set holds f is_sequence_on G implies (f -: p) is_sequence_on G

    proof

      let D be non empty set;

      let f be FinSequence of D;

      let G be Matrix of D;

      let p be set;

      assume f is_sequence_on G;

      then (f | (p .. f)) is_sequence_on G by GOBOARD1: 22;

      hence thesis by FINSEQ_5:def 1;

    end;

    theorem :: JORDAN1G:3

    

     Th3: for D be non empty set holds for f be FinSequence of D holds for G be Matrix of D holds for p be Element of D st p in ( rng f) holds f is_sequence_on G implies (f :- p) is_sequence_on G

    proof

      let D be non empty set;

      let f be FinSequence of D;

      let G be Matrix of D;

      let p be Element of D;

      assume that

       A1: p in ( rng f) and

       A2: f is_sequence_on G;

      ex i be Element of NAT st (i + 1) = (p .. f) & (f :- p) = (f /^ i) by A1, FINSEQ_5: 49;

      hence thesis by A2, JORDAN8: 2;

    end;

    theorem :: JORDAN1G:4

    

     Th4: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds ( Upper_Seq (C,n)) is_sequence_on ( Gauge (C,n))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      ( Cage (C,n)) is_sequence_on ( Gauge (C,n)) by JORDAN9:def 1;

      then ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) is_sequence_on ( Gauge (C,n)) by REVROT_1: 34;

      then (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) -: ( E-max ( L~ ( Cage (C,n))))) is_sequence_on ( Gauge (C,n)) by Th2;

      hence thesis by JORDAN1E:def 1;

    end;

    theorem :: JORDAN1G:5

    

     Th5: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds ( Lower_Seq (C,n)) is_sequence_on ( Gauge (C,n))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      ( Cage (C,n)) is_sequence_on ( Gauge (C,n)) by JORDAN9:def 1;

      then

       A1: ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) is_sequence_on ( Gauge (C,n)) by REVROT_1: 34;

      ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 46;

      then ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by FINSEQ_6: 90, SPRECT_2: 43;

      then (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) :- ( E-max ( L~ ( Cage (C,n))))) is_sequence_on ( Gauge (C,n)) by A1, Th3;

      hence thesis by JORDAN1E:def 2;

    end;

    registration

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let n be Nat;

      cluster ( Upper_Seq (C,n)) -> standard;

      coherence

      proof

        ( Upper_Seq (C,n)) is_sequence_on ( Gauge (C,n)) by Th4;

        hence thesis by JORDAN8: 4;

      end;

      cluster ( Lower_Seq (C,n)) -> standard;

      coherence

      proof

        ( Lower_Seq (C,n)) is_sequence_on ( Gauge (C,n)) by Th5;

        hence thesis by JORDAN8: 4;

      end;

    end

    theorem :: JORDAN1G:6

    

     Th6: for G be Y_equal-in-column Y_increasing-in-line Matrix of ( TOP-REAL 2) holds for i1,i2,j1,j2 be Nat st [i1, j1] in ( Indices G) & [i2, j2] in ( Indices G) holds ((G * (i1,j1)) `2 ) = ((G * (i2,j2)) `2 ) implies j1 = j2

    proof

      let G be Y_equal-in-column Y_increasing-in-line Matrix of ( TOP-REAL 2);

      let i1,i2,j1,j2 be Nat;

      assume that

       A1: [i1, j1] in ( Indices G) and

       A2: [i2, j2] in ( Indices G) and

       A3: ((G * (i1,j1)) `2 ) = ((G * (i2,j2)) `2 ) and

       A4: j1 <> j2;

      

       A5: 1 <= j1 & j1 <= ( width G) by A1, MATRIX_0: 32;

      

       A6: j1 < j2 or j1 > j2 by A4, XXREAL_0: 1;

      

       A7: 1 <= i2 & i2 <= ( len G) by A2, MATRIX_0: 32;

      

       A8: 1 <= j2 & j2 <= ( width G) by A2, MATRIX_0: 32;

      

       A9: 1 <= i1 & i1 <= ( len G) by A1, MATRIX_0: 32;

      

      then ((G * (i1,j2)) `2 ) = ((G * (1,j2)) `2 ) by A8, GOBOARD5: 1

      .= ((G * (i2,j2)) `2 ) by A7, A8, GOBOARD5: 1;

      hence contradiction by A3, A9, A5, A8, A6, GOBOARD5: 4;

    end;

    theorem :: JORDAN1G:7

    

     Th7: for G be X_equal-in-line X_increasing-in-column Matrix of ( TOP-REAL 2) holds for i1,i2,j1,j2 be Nat st [i1, j1] in ( Indices G) & [i2, j2] in ( Indices G) holds ((G * (i1,j1)) `1 ) = ((G * (i2,j2)) `1 ) implies i1 = i2

    proof

      let G be X_equal-in-line X_increasing-in-column Matrix of ( TOP-REAL 2);

      let i1,i2,j1,j2 be Nat;

      assume that

       A1: [i1, j1] in ( Indices G) and

       A2: [i2, j2] in ( Indices G) and

       A3: ((G * (i1,j1)) `1 ) = ((G * (i2,j2)) `1 ) and

       A4: i1 <> i2;

      

       A5: 1 <= i1 & i1 <= ( len G) by A1, MATRIX_0: 32;

      

       A6: 1 <= i2 & i2 <= ( len G) by A2, MATRIX_0: 32;

      

       A7: 1 <= j2 & j2 <= ( width G) by A2, MATRIX_0: 32;

      

       A8: i1 < i2 or i1 > i2 by A4, XXREAL_0: 1;

      1 <= j1 & j1 <= ( width G) by A1, MATRIX_0: 32;

      

      then ((G * (i1,j1)) `1 ) = ((G * (i1,1)) `1 ) by A5, GOBOARD5: 2

      .= ((G * (i1,j2)) `1 ) by A5, A7, GOBOARD5: 2;

      hence contradiction by A3, A5, A6, A7, A8, GOBOARD5: 3;

    end;

    theorem :: JORDAN1G:8

    

     Th8: for f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2) st ((f /. 1) <> ( N-min ( L~ f)) & (f /. ( len f)) <> ( N-min ( L~ f))) or ((f /. 1) <> ( N-max ( L~ f)) & (f /. ( len f)) <> ( N-max ( L~ f))) holds (( N-min ( L~ f)) `1 ) < (( N-max ( L~ f)) `1 )

    proof

      let f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2);

      set p = ( N-min ( L~ f));

      set i = (p .. f);

      assume

       A1: (f /. 1) <> ( N-min ( L~ f)) & (f /. ( len f)) <> ( N-min ( L~ f)) or (f /. 1) <> ( N-max ( L~ f)) & (f /. ( len f)) <> ( N-max ( L~ f));

      

       A2: ( len f) >= 2 by NAT_D: 60;

      

       A3: (p `2 ) = ( N-bound ( L~ f)) by EUCLID: 52;

      

       A4: p in ( rng f) by SPRECT_2: 39;

      then

       A5: i in ( dom f) by FINSEQ_4: 20;

      then

       A6: 1 <= i & i <= ( len f) by FINSEQ_3: 25;

      

       A7: p = (f . i) by A4, FINSEQ_4: 19

      .= (f /. i) by A5, PARTFUN1:def 6;

      per cases by A6, XXREAL_0: 1;

        suppose

         A8: i = 1;

        (p `2 ) = (( N-max ( L~ f)) `2 ) by PSCOMP_1: 37;

        then

         A9: (p `1 ) <> (( N-max ( L~ f)) `1 ) by A1, A7, A8, TOPREAL3: 6;

        (p `1 ) <= (( N-max ( L~ f)) `1 ) by PSCOMP_1: 38;

        hence thesis by A9, XXREAL_0: 1;

      end;

        suppose

         A10: i = ( len f);

        (p `2 ) = (( N-max ( L~ f)) `2 ) by PSCOMP_1: 37;

        then

         A11: (p `1 ) <> (( N-max ( L~ f)) `1 ) by A1, A7, A10, TOPREAL3: 6;

        (p `1 ) <= (( N-max ( L~ f)) `1 ) by PSCOMP_1: 38;

        hence thesis by A11, XXREAL_0: 1;

      end;

        suppose that

         A12: 1 < i and

         A13: i < ( len f);

        

         A14: ((i -' 1) + 1) = i by A12, XREAL_1: 235;

        then

         A15: (i -' 1) >= 1 by A12, NAT_1: 13;

        then

         A16: (f /. (i -' 1)) in ( LSeg (f,(i -' 1))) by A13, A14, TOPREAL1: 21;

        (i -' 1) <= i by A14, NAT_1: 11;

        then (i -' 1) <= ( len f) by A13, XXREAL_0: 2;

        then

         A17: (i -' 1) in ( dom f) by A15, FINSEQ_3: 25;

        then

         A18: (f /. (i -' 1)) in ( L~ f) by A2, GOBOARD1: 1;

        

         A19: (i + 1) <= ( len f) by A13, NAT_1: 13;

        then

         A20: (f /. (i + 1)) in ( LSeg (f,i)) by A12, TOPREAL1: 21;

        (i + 1) >= 1 by NAT_1: 11;

        then

         A21: (i + 1) in ( dom f) by A19, FINSEQ_3: 25;

        then

         A22: (f /. (i + 1)) in ( L~ f) by A2, GOBOARD1: 1;

        

         A23: p <> (f /. (i + 1)) by A4, A7, A21, FINSEQ_4: 20, GOBOARD7: 29;

        

         A24: p in ( LSeg (f,i)) by A7, A12, A19, TOPREAL1: 21;

        

         A25: p in ( LSeg (f,(i -' 1))) by A7, A13, A14, A15, TOPREAL1: 21;

        

         A26: p <> (f /. (i -' 1)) by A5, A7, A14, A17, GOBOARD7: 29;

        

         A27: not (( LSeg (f,(i -' 1))) is vertical & ( LSeg (f,i)) is vertical)

        proof

          assume ( LSeg (f,(i -' 1))) is vertical & ( LSeg (f,i)) is vertical;

          then

           A28: (p `1 ) = ((f /. (i + 1)) `1 ) & (p `1 ) = ((f /. (i -' 1)) `1 ) by A25, A24, A16, A20, SPPOL_1:def 3;

          

           A29: ((f /. (i + 1)) `2 ) <= ((f /. (i -' 1)) `2 ) or ((f /. (i + 1)) `2 ) >= ((f /. (i -' 1)) `2 );

          

           A30: (p `2 ) >= ((f /. (i + 1)) `2 ) & (p `2 ) >= ((f /. (i -' 1)) `2 ) by A3, A18, A22, PSCOMP_1: 24;

          ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) & ( LSeg (f,(i -' 1))) = ( LSeg ((f /. i),(f /. (i -' 1)))) by A12, A13, A14, A15, A19, TOPREAL1:def 3;

          then (f /. (i -' 1)) in ( LSeg (f,i)) or (f /. (i + 1)) in ( LSeg (f,(i -' 1))) by A7, A28, A30, A29, GOBOARD7: 7;

          then (f /. (i -' 1)) in (( LSeg (f,(i -' 1))) /\ ( LSeg (f,i))) or (f /. (i + 1)) in (( LSeg (f,(i -' 1))) /\ ( LSeg (f,i))) by A16, A20, XBOOLE_0:def 4;

          then (((i -' 1) + 1) + 1) = ((i -' 1) + (1 + 1)) & (( LSeg (f,(i -' 1))) /\ ( LSeg (f,i))) <> {(f /. i)} by A7, A26, A23, TARSKI:def 1;

          hence contradiction by A14, A15, A19, TOPREAL1:def 6;

        end;

        now

          per cases by A27, SPPOL_1: 19;

            suppose ( LSeg (f,(i -' 1))) is horizontal;

            then

             A31: (p `2 ) = ((f /. (i -' 1)) `2 ) by A25, A16, SPPOL_1:def 2;

            then

             A32: (f /. (i -' 1)) in ( N-most ( L~ f)) by A2, A3, A17, GOBOARD1: 1, SPRECT_2: 10;

            then

             A33: ((f /. (i -' 1)) `1 ) >= (p `1 ) by PSCOMP_1: 39;

            ((f /. (i -' 1)) `1 ) <> (p `1 ) by A5, A7, A14, A17, A31, GOBOARD7: 29, TOPREAL3: 6;

            then

             A34: ((f /. (i -' 1)) `1 ) > (p `1 ) by A33, XXREAL_0: 1;

            ((f /. (i -' 1)) `1 ) <= (( N-max ( L~ f)) `1 ) by A32, PSCOMP_1: 39;

            hence thesis by A34, XXREAL_0: 2;

          end;

            suppose ( LSeg (f,i)) is horizontal;

            then

             A35: (p `2 ) = ((f /. (i + 1)) `2 ) by A24, A20, SPPOL_1:def 2;

            then

             A36: (f /. (i + 1)) in ( N-most ( L~ f)) by A2, A3, A21, GOBOARD1: 1, SPRECT_2: 10;

            then

             A37: ((f /. (i + 1)) `1 ) >= (p `1 ) by PSCOMP_1: 39;

            ((f /. (i + 1)) `1 ) <> (p `1 ) by A5, A7, A21, A35, GOBOARD7: 29, TOPREAL3: 6;

            then

             A38: ((f /. (i + 1)) `1 ) > (p `1 ) by A37, XXREAL_0: 1;

            ((f /. (i + 1)) `1 ) <= (( N-max ( L~ f)) `1 ) by A36, PSCOMP_1: 39;

            hence thesis by A38, XXREAL_0: 2;

          end;

        end;

        hence thesis;

      end;

    end;

    theorem :: JORDAN1G:9

    for f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2) st ((f /. 1) <> ( N-min ( L~ f)) & (f /. ( len f)) <> ( N-min ( L~ f))) or ((f /. 1) <> ( N-max ( L~ f)) & (f /. ( len f)) <> ( N-max ( L~ f))) holds ( N-min ( L~ f)) <> ( N-max ( L~ f))

    proof

      let f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2);

      assume (f /. 1) <> ( N-min ( L~ f)) & (f /. ( len f)) <> ( N-min ( L~ f)) or (f /. 1) <> ( N-max ( L~ f)) & (f /. ( len f)) <> ( N-max ( L~ f));

      then (( N-min ( L~ f)) `1 ) < (( N-max ( L~ f)) `1 ) by Th8;

      hence thesis;

    end;

    theorem :: JORDAN1G:10

    

     Th10: for f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2) st ((f /. 1) <> ( S-min ( L~ f)) & (f /. ( len f)) <> ( S-min ( L~ f))) or ((f /. 1) <> ( S-max ( L~ f)) & (f /. ( len f)) <> ( S-max ( L~ f))) holds (( S-min ( L~ f)) `1 ) < (( S-max ( L~ f)) `1 )

    proof

      let f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2);

      set p = ( S-min ( L~ f));

      set i = (p .. f);

      assume

       A1: (f /. 1) <> ( S-min ( L~ f)) & (f /. ( len f)) <> ( S-min ( L~ f)) or (f /. 1) <> ( S-max ( L~ f)) & (f /. ( len f)) <> ( S-max ( L~ f));

      

       A2: ( len f) >= 2 by NAT_D: 60;

      

       A3: (p `2 ) = ( S-bound ( L~ f)) by EUCLID: 52;

      

       A4: p in ( rng f) by SPRECT_2: 41;

      then

       A5: i in ( dom f) by FINSEQ_4: 20;

      then

       A6: 1 <= i & i <= ( len f) by FINSEQ_3: 25;

      

       A7: p = (f . i) by A4, FINSEQ_4: 19

      .= (f /. i) by A5, PARTFUN1:def 6;

      per cases by A6, XXREAL_0: 1;

        suppose

         A8: i = 1;

        (p `2 ) = (( S-max ( L~ f)) `2 ) by PSCOMP_1: 53;

        then

         A9: (p `1 ) <> (( S-max ( L~ f)) `1 ) by A1, A7, A8, TOPREAL3: 6;

        (p `1 ) <= (( S-max ( L~ f)) `1 ) by PSCOMP_1: 54;

        hence thesis by A9, XXREAL_0: 1;

      end;

        suppose

         A10: i = ( len f);

        (p `2 ) = (( S-max ( L~ f)) `2 ) by PSCOMP_1: 53;

        then

         A11: (p `1 ) <> (( S-max ( L~ f)) `1 ) by A1, A7, A10, TOPREAL3: 6;

        (p `1 ) <= (( S-max ( L~ f)) `1 ) by PSCOMP_1: 54;

        hence thesis by A11, XXREAL_0: 1;

      end;

        suppose that

         A12: 1 < i and

         A13: i < ( len f);

        

         A14: ((i -' 1) + 1) = i by A12, XREAL_1: 235;

        then

         A15: (i -' 1) >= 1 by A12, NAT_1: 13;

        then

         A16: (f /. (i -' 1)) in ( LSeg (f,(i -' 1))) by A13, A14, TOPREAL1: 21;

        (i -' 1) <= i by A14, NAT_1: 11;

        then (i -' 1) <= ( len f) by A13, XXREAL_0: 2;

        then

         A17: (i -' 1) in ( dom f) by A15, FINSEQ_3: 25;

        then

         A18: (f /. (i -' 1)) in ( L~ f) by A2, GOBOARD1: 1;

        

         A19: (i + 1) <= ( len f) by A13, NAT_1: 13;

        then

         A20: (f /. (i + 1)) in ( LSeg (f,i)) by A12, TOPREAL1: 21;

        (i + 1) >= 1 by NAT_1: 11;

        then

         A21: (i + 1) in ( dom f) by A19, FINSEQ_3: 25;

        then

         A22: (f /. (i + 1)) in ( L~ f) by A2, GOBOARD1: 1;

        

         A23: p <> (f /. (i + 1)) by A4, A7, A21, FINSEQ_4: 20, GOBOARD7: 29;

        

         A24: p in ( LSeg (f,i)) by A7, A12, A19, TOPREAL1: 21;

        

         A25: p in ( LSeg (f,(i -' 1))) by A7, A13, A14, A15, TOPREAL1: 21;

        

         A26: p <> (f /. (i -' 1)) by A5, A7, A14, A17, GOBOARD7: 29;

        

         A27: not (( LSeg (f,(i -' 1))) is vertical & ( LSeg (f,i)) is vertical)

        proof

          assume ( LSeg (f,(i -' 1))) is vertical & ( LSeg (f,i)) is vertical;

          then

           A28: (p `1 ) = ((f /. (i + 1)) `1 ) & (p `1 ) = ((f /. (i -' 1)) `1 ) by A25, A24, A16, A20, SPPOL_1:def 3;

          

           A29: ((f /. (i + 1)) `2 ) <= ((f /. (i -' 1)) `2 ) or ((f /. (i + 1)) `2 ) >= ((f /. (i -' 1)) `2 );

          

           A30: (p `2 ) <= ((f /. (i + 1)) `2 ) & (p `2 ) <= ((f /. (i -' 1)) `2 ) by A3, A18, A22, PSCOMP_1: 24;

          ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) & ( LSeg (f,(i -' 1))) = ( LSeg ((f /. i),(f /. (i -' 1)))) by A12, A13, A14, A15, A19, TOPREAL1:def 3;

          then (f /. (i -' 1)) in ( LSeg (f,i)) or (f /. (i + 1)) in ( LSeg (f,(i -' 1))) by A7, A28, A30, A29, GOBOARD7: 7;

          then (f /. (i -' 1)) in (( LSeg (f,(i -' 1))) /\ ( LSeg (f,i))) or (f /. (i + 1)) in (( LSeg (f,(i -' 1))) /\ ( LSeg (f,i))) by A16, A20, XBOOLE_0:def 4;

          then (((i -' 1) + 1) + 1) = ((i -' 1) + (1 + 1)) & (( LSeg (f,(i -' 1))) /\ ( LSeg (f,i))) <> {(f /. i)} by A7, A26, A23, TARSKI:def 1;

          hence contradiction by A14, A15, A19, TOPREAL1:def 6;

        end;

        now

          per cases by A27, SPPOL_1: 19;

            suppose ( LSeg (f,(i -' 1))) is horizontal;

            then

             A31: (p `2 ) = ((f /. (i -' 1)) `2 ) by A25, A16, SPPOL_1:def 2;

            then

             A32: (f /. (i -' 1)) in ( S-most ( L~ f)) by A2, A3, A17, GOBOARD1: 1, SPRECT_2: 11;

            then

             A33: ((f /. (i -' 1)) `1 ) >= (p `1 ) by PSCOMP_1: 55;

            ((f /. (i -' 1)) `1 ) <> (p `1 ) by A5, A7, A14, A17, A31, GOBOARD7: 29, TOPREAL3: 6;

            then

             A34: ((f /. (i -' 1)) `1 ) > (p `1 ) by A33, XXREAL_0: 1;

            ((f /. (i -' 1)) `1 ) <= (( S-max ( L~ f)) `1 ) by A32, PSCOMP_1: 55;

            hence thesis by A34, XXREAL_0: 2;

          end;

            suppose ( LSeg (f,i)) is horizontal;

            then

             A35: (p `2 ) = ((f /. (i + 1)) `2 ) by A24, A20, SPPOL_1:def 2;

            then

             A36: (f /. (i + 1)) in ( S-most ( L~ f)) by A2, A3, A21, GOBOARD1: 1, SPRECT_2: 11;

            then

             A37: ((f /. (i + 1)) `1 ) >= (p `1 ) by PSCOMP_1: 55;

            ((f /. (i + 1)) `1 ) <> (p `1 ) by A5, A7, A21, A35, GOBOARD7: 29, TOPREAL3: 6;

            then

             A38: ((f /. (i + 1)) `1 ) > (p `1 ) by A37, XXREAL_0: 1;

            ((f /. (i + 1)) `1 ) <= (( S-max ( L~ f)) `1 ) by A36, PSCOMP_1: 55;

            hence thesis by A38, XXREAL_0: 2;

          end;

        end;

        hence thesis;

      end;

    end;

    theorem :: JORDAN1G:11

    for f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2) st ((f /. 1) <> ( S-min ( L~ f)) & (f /. ( len f)) <> ( S-min ( L~ f))) or ((f /. 1) <> ( S-max ( L~ f)) & (f /. ( len f)) <> ( S-max ( L~ f))) holds ( S-min ( L~ f)) <> ( S-max ( L~ f))

    proof

      let f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2);

      assume (f /. 1) <> ( S-min ( L~ f)) & (f /. ( len f)) <> ( S-min ( L~ f)) or (f /. 1) <> ( S-max ( L~ f)) & (f /. ( len f)) <> ( S-max ( L~ f));

      then (( S-min ( L~ f)) `1 ) < (( S-max ( L~ f)) `1 ) by Th10;

      hence thesis;

    end;

    theorem :: JORDAN1G:12

    

     Th12: for f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2) st ((f /. 1) <> ( W-min ( L~ f)) & (f /. ( len f)) <> ( W-min ( L~ f))) or ((f /. 1) <> ( W-max ( L~ f)) & (f /. ( len f)) <> ( W-max ( L~ f))) holds (( W-min ( L~ f)) `2 ) < (( W-max ( L~ f)) `2 )

    proof

      let f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2);

      set p = ( W-min ( L~ f));

      set i = (p .. f);

      assume

       A1: (f /. 1) <> ( W-min ( L~ f)) & (f /. ( len f)) <> ( W-min ( L~ f)) or (f /. 1) <> ( W-max ( L~ f)) & (f /. ( len f)) <> ( W-max ( L~ f));

      

       A2: ( len f) >= 2 by NAT_D: 60;

      

       A3: (p `1 ) = ( W-bound ( L~ f)) by EUCLID: 52;

      

       A4: p in ( rng f) by SPRECT_2: 43;

      then

       A5: i in ( dom f) by FINSEQ_4: 20;

      then

       A6: 1 <= i & i <= ( len f) by FINSEQ_3: 25;

      

       A7: p = (f . i) by A4, FINSEQ_4: 19

      .= (f /. i) by A5, PARTFUN1:def 6;

      per cases by A6, XXREAL_0: 1;

        suppose

         A8: i = 1;

        (p `1 ) = (( W-max ( L~ f)) `1 ) by PSCOMP_1: 29;

        then

         A9: (p `2 ) <> (( W-max ( L~ f)) `2 ) by A1, A7, A8, TOPREAL3: 6;

        (p `2 ) <= (( W-max ( L~ f)) `2 ) by PSCOMP_1: 30;

        hence thesis by A9, XXREAL_0: 1;

      end;

        suppose

         A10: i = ( len f);

        (p `1 ) = (( W-max ( L~ f)) `1 ) by PSCOMP_1: 29;

        then

         A11: (p `2 ) <> (( W-max ( L~ f)) `2 ) by A1, A7, A10, TOPREAL3: 6;

        (p `2 ) <= (( W-max ( L~ f)) `2 ) by PSCOMP_1: 30;

        hence thesis by A11, XXREAL_0: 1;

      end;

        suppose that

         A12: 1 < i and

         A13: i < ( len f);

        

         A14: ((i -' 1) + 1) = i by A12, XREAL_1: 235;

        then

         A15: (i -' 1) >= 1 by A12, NAT_1: 13;

        then

         A16: (f /. (i -' 1)) in ( LSeg (f,(i -' 1))) by A13, A14, TOPREAL1: 21;

        (i -' 1) <= i by A14, NAT_1: 11;

        then (i -' 1) <= ( len f) by A13, XXREAL_0: 2;

        then

         A17: (i -' 1) in ( dom f) by A15, FINSEQ_3: 25;

        then

         A18: (f /. (i -' 1)) in ( L~ f) by A2, GOBOARD1: 1;

        

         A19: (i + 1) <= ( len f) by A13, NAT_1: 13;

        then

         A20: (f /. (i + 1)) in ( LSeg (f,i)) by A12, TOPREAL1: 21;

        (i + 1) >= 1 by NAT_1: 11;

        then

         A21: (i + 1) in ( dom f) by A19, FINSEQ_3: 25;

        then

         A22: (f /. (i + 1)) in ( L~ f) by A2, GOBOARD1: 1;

        

         A23: p <> (f /. (i + 1)) by A4, A7, A21, FINSEQ_4: 20, GOBOARD7: 29;

        

         A24: p in ( LSeg (f,i)) by A7, A12, A19, TOPREAL1: 21;

        

         A25: p in ( LSeg (f,(i -' 1))) by A7, A13, A14, A15, TOPREAL1: 21;

        

         A26: p <> (f /. (i -' 1)) by A5, A7, A14, A17, GOBOARD7: 29;

        

         A27: not (( LSeg (f,(i -' 1))) is horizontal & ( LSeg (f,i)) is horizontal)

        proof

          assume ( LSeg (f,(i -' 1))) is horizontal & ( LSeg (f,i)) is horizontal;

          then

           A28: (p `2 ) = ((f /. (i + 1)) `2 ) & (p `2 ) = ((f /. (i -' 1)) `2 ) by A25, A24, A16, A20, SPPOL_1:def 2;

          

           A29: ((f /. (i + 1)) `1 ) <= ((f /. (i -' 1)) `1 ) or ((f /. (i + 1)) `1 ) >= ((f /. (i -' 1)) `1 );

          

           A30: (p `1 ) <= ((f /. (i + 1)) `1 ) & (p `1 ) <= ((f /. (i -' 1)) `1 ) by A3, A18, A22, PSCOMP_1: 24;

          ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) & ( LSeg (f,(i -' 1))) = ( LSeg ((f /. i),(f /. (i -' 1)))) by A12, A13, A14, A15, A19, TOPREAL1:def 3;

          then (f /. (i -' 1)) in ( LSeg (f,i)) or (f /. (i + 1)) in ( LSeg (f,(i -' 1))) by A7, A28, A30, A29, GOBOARD7: 8;

          then (f /. (i -' 1)) in (( LSeg (f,(i -' 1))) /\ ( LSeg (f,i))) or (f /. (i + 1)) in (( LSeg (f,(i -' 1))) /\ ( LSeg (f,i))) by A16, A20, XBOOLE_0:def 4;

          then (((i -' 1) + 1) + 1) = ((i -' 1) + (1 + 1)) & (( LSeg (f,(i -' 1))) /\ ( LSeg (f,i))) <> {(f /. i)} by A7, A26, A23, TARSKI:def 1;

          hence contradiction by A14, A15, A19, TOPREAL1:def 6;

        end;

        now

          per cases by A27, SPPOL_1: 19;

            suppose ( LSeg (f,(i -' 1))) is vertical;

            then

             A31: (p `1 ) = ((f /. (i -' 1)) `1 ) by A25, A16, SPPOL_1:def 3;

            then

             A32: (f /. (i -' 1)) in ( W-most ( L~ f)) by A2, A3, A17, GOBOARD1: 1, SPRECT_2: 12;

            then

             A33: ((f /. (i -' 1)) `2 ) >= (p `2 ) by PSCOMP_1: 31;

            ((f /. (i -' 1)) `2 ) <> (p `2 ) by A5, A7, A14, A17, A31, GOBOARD7: 29, TOPREAL3: 6;

            then

             A34: ((f /. (i -' 1)) `2 ) > (p `2 ) by A33, XXREAL_0: 1;

            ((f /. (i -' 1)) `2 ) <= (( W-max ( L~ f)) `2 ) by A32, PSCOMP_1: 31;

            hence thesis by A34, XXREAL_0: 2;

          end;

            suppose ( LSeg (f,i)) is vertical;

            then

             A35: (p `1 ) = ((f /. (i + 1)) `1 ) by A24, A20, SPPOL_1:def 3;

            then

             A36: (f /. (i + 1)) in ( W-most ( L~ f)) by A2, A3, A21, GOBOARD1: 1, SPRECT_2: 12;

            then

             A37: ((f /. (i + 1)) `2 ) >= (p `2 ) by PSCOMP_1: 31;

            ((f /. (i + 1)) `2 ) <> (p `2 ) by A5, A7, A21, A35, GOBOARD7: 29, TOPREAL3: 6;

            then

             A38: ((f /. (i + 1)) `2 ) > (p `2 ) by A37, XXREAL_0: 1;

            ((f /. (i + 1)) `2 ) <= (( W-max ( L~ f)) `2 ) by A36, PSCOMP_1: 31;

            hence thesis by A38, XXREAL_0: 2;

          end;

        end;

        hence thesis;

      end;

    end;

    theorem :: JORDAN1G:13

    for f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2) st ((f /. 1) <> ( W-min ( L~ f)) & (f /. ( len f)) <> ( W-min ( L~ f))) or ((f /. 1) <> ( W-max ( L~ f)) & (f /. ( len f)) <> ( W-max ( L~ f))) holds ( W-min ( L~ f)) <> ( W-max ( L~ f))

    proof

      let f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2);

      assume (f /. 1) <> ( W-min ( L~ f)) & (f /. ( len f)) <> ( W-min ( L~ f)) or (f /. 1) <> ( W-max ( L~ f)) & (f /. ( len f)) <> ( W-max ( L~ f));

      then (( W-min ( L~ f)) `2 ) < (( W-max ( L~ f)) `2 ) by Th12;

      hence thesis;

    end;

    theorem :: JORDAN1G:14

    

     Th14: for f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2) st ((f /. 1) <> ( E-min ( L~ f)) & (f /. ( len f)) <> ( E-min ( L~ f))) or ((f /. 1) <> ( E-max ( L~ f)) & (f /. ( len f)) <> ( E-max ( L~ f))) holds (( E-min ( L~ f)) `2 ) < (( E-max ( L~ f)) `2 )

    proof

      let f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2);

      set p = ( E-min ( L~ f));

      set i = (p .. f);

      assume

       A1: (f /. 1) <> ( E-min ( L~ f)) & (f /. ( len f)) <> ( E-min ( L~ f)) or (f /. 1) <> ( E-max ( L~ f)) & (f /. ( len f)) <> ( E-max ( L~ f));

      

       A2: ( len f) >= 2 by NAT_D: 60;

      

       A3: (p `1 ) = ( E-bound ( L~ f)) by EUCLID: 52;

      

       A4: p in ( rng f) by SPRECT_2: 45;

      then

       A5: i in ( dom f) by FINSEQ_4: 20;

      then

       A6: 1 <= i & i <= ( len f) by FINSEQ_3: 25;

      

       A7: p = (f . i) by A4, FINSEQ_4: 19

      .= (f /. i) by A5, PARTFUN1:def 6;

      per cases by A6, XXREAL_0: 1;

        suppose

         A8: i = 1;

        (p `1 ) = (( E-max ( L~ f)) `1 ) by PSCOMP_1: 45;

        then

         A9: (p `2 ) <> (( E-max ( L~ f)) `2 ) by A1, A7, A8, TOPREAL3: 6;

        (p `2 ) <= (( E-max ( L~ f)) `2 ) by PSCOMP_1: 46;

        hence thesis by A9, XXREAL_0: 1;

      end;

        suppose

         A10: i = ( len f);

        (p `1 ) = (( E-max ( L~ f)) `1 ) by PSCOMP_1: 45;

        then

         A11: (p `2 ) <> (( E-max ( L~ f)) `2 ) by A1, A7, A10, TOPREAL3: 6;

        (p `2 ) <= (( E-max ( L~ f)) `2 ) by PSCOMP_1: 46;

        hence thesis by A11, XXREAL_0: 1;

      end;

        suppose that

         A12: 1 < i and

         A13: i < ( len f);

        

         A14: ((i -' 1) + 1) = i by A12, XREAL_1: 235;

        then

         A15: (i -' 1) >= 1 by A12, NAT_1: 13;

        then

         A16: (f /. (i -' 1)) in ( LSeg (f,(i -' 1))) by A13, A14, TOPREAL1: 21;

        (i -' 1) <= i by A14, NAT_1: 11;

        then (i -' 1) <= ( len f) by A13, XXREAL_0: 2;

        then

         A17: (i -' 1) in ( dom f) by A15, FINSEQ_3: 25;

        then

         A18: (f /. (i -' 1)) in ( L~ f) by A2, GOBOARD1: 1;

        

         A19: (i + 1) <= ( len f) by A13, NAT_1: 13;

        then

         A20: (f /. (i + 1)) in ( LSeg (f,i)) by A12, TOPREAL1: 21;

        (i + 1) >= 1 by NAT_1: 11;

        then

         A21: (i + 1) in ( dom f) by A19, FINSEQ_3: 25;

        then

         A22: (f /. (i + 1)) in ( L~ f) by A2, GOBOARD1: 1;

        

         A23: p <> (f /. (i + 1)) by A4, A7, A21, FINSEQ_4: 20, GOBOARD7: 29;

        

         A24: p in ( LSeg (f,i)) by A7, A12, A19, TOPREAL1: 21;

        

         A25: p in ( LSeg (f,(i -' 1))) by A7, A13, A14, A15, TOPREAL1: 21;

        

         A26: p <> (f /. (i -' 1)) by A5, A7, A14, A17, GOBOARD7: 29;

        

         A27: not (( LSeg (f,(i -' 1))) is horizontal & ( LSeg (f,i)) is horizontal)

        proof

          assume ( LSeg (f,(i -' 1))) is horizontal & ( LSeg (f,i)) is horizontal;

          then

           A28: (p `2 ) = ((f /. (i + 1)) `2 ) & (p `2 ) = ((f /. (i -' 1)) `2 ) by A25, A24, A16, A20, SPPOL_1:def 2;

          

           A29: ((f /. (i + 1)) `1 ) <= ((f /. (i -' 1)) `1 ) or ((f /. (i + 1)) `1 ) >= ((f /. (i -' 1)) `1 );

          

           A30: (p `1 ) >= ((f /. (i + 1)) `1 ) & (p `1 ) >= ((f /. (i -' 1)) `1 ) by A3, A18, A22, PSCOMP_1: 24;

          ( LSeg (f,i)) = ( LSeg ((f /. i),(f /. (i + 1)))) & ( LSeg (f,(i -' 1))) = ( LSeg ((f /. i),(f /. (i -' 1)))) by A12, A13, A14, A15, A19, TOPREAL1:def 3;

          then (f /. (i -' 1)) in ( LSeg (f,i)) or (f /. (i + 1)) in ( LSeg (f,(i -' 1))) by A7, A28, A30, A29, GOBOARD7: 8;

          then (f /. (i -' 1)) in (( LSeg (f,(i -' 1))) /\ ( LSeg (f,i))) or (f /. (i + 1)) in (( LSeg (f,(i -' 1))) /\ ( LSeg (f,i))) by A16, A20, XBOOLE_0:def 4;

          then (((i -' 1) + 1) + 1) = ((i -' 1) + (1 + 1)) & (( LSeg (f,(i -' 1))) /\ ( LSeg (f,i))) <> {(f /. i)} by A7, A26, A23, TARSKI:def 1;

          hence contradiction by A14, A15, A19, TOPREAL1:def 6;

        end;

        now

          per cases by A27, SPPOL_1: 19;

            suppose ( LSeg (f,(i -' 1))) is vertical;

            then

             A31: (p `1 ) = ((f /. (i -' 1)) `1 ) by A25, A16, SPPOL_1:def 3;

            then

             A32: (f /. (i -' 1)) in ( E-most ( L~ f)) by A2, A3, A17, GOBOARD1: 1, SPRECT_2: 13;

            then

             A33: ((f /. (i -' 1)) `2 ) >= (p `2 ) by PSCOMP_1: 47;

            ((f /. (i -' 1)) `2 ) <> (p `2 ) by A5, A7, A14, A17, A31, GOBOARD7: 29, TOPREAL3: 6;

            then

             A34: ((f /. (i -' 1)) `2 ) > (p `2 ) by A33, XXREAL_0: 1;

            ((f /. (i -' 1)) `2 ) <= (( E-max ( L~ f)) `2 ) by A32, PSCOMP_1: 47;

            hence thesis by A34, XXREAL_0: 2;

          end;

            suppose ( LSeg (f,i)) is vertical;

            then

             A35: (p `1 ) = ((f /. (i + 1)) `1 ) by A24, A20, SPPOL_1:def 3;

            then

             A36: (f /. (i + 1)) in ( E-most ( L~ f)) by A2, A3, A21, GOBOARD1: 1, SPRECT_2: 13;

            then

             A37: ((f /. (i + 1)) `2 ) >= (p `2 ) by PSCOMP_1: 47;

            ((f /. (i + 1)) `2 ) <> (p `2 ) by A5, A7, A21, A35, GOBOARD7: 29, TOPREAL3: 6;

            then

             A38: ((f /. (i + 1)) `2 ) > (p `2 ) by A37, XXREAL_0: 1;

            ((f /. (i + 1)) `2 ) <= (( E-max ( L~ f)) `2 ) by A36, PSCOMP_1: 47;

            hence thesis by A38, XXREAL_0: 2;

          end;

        end;

        hence thesis;

      end;

    end;

    theorem :: JORDAN1G:15

    for f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2) st ((f /. 1) <> ( E-min ( L~ f)) & (f /. ( len f)) <> ( E-min ( L~ f))) or ((f /. 1) <> ( E-max ( L~ f)) & (f /. ( len f)) <> ( E-max ( L~ f))) holds ( E-min ( L~ f)) <> ( E-max ( L~ f))

    proof

      let f be standard special unfolded non trivial FinSequence of ( TOP-REAL 2);

      assume (f /. 1) <> ( E-min ( L~ f)) & (f /. ( len f)) <> ( E-min ( L~ f)) or (f /. 1) <> ( E-max ( L~ f)) & (f /. ( len f)) <> ( E-max ( L~ f));

      then (( E-min ( L~ f)) `2 ) < (( E-max ( L~ f)) `2 ) by Th14;

      hence thesis;

    end;

    theorem :: JORDAN1G:16

    

     Th16: for D be non empty set holds for f be FinSequence of D holds for p,q be Element of D st p in ( rng f) & q in ( rng f) & (q .. f) <= (p .. f) holds ((f -: p) :- q) = ((f :- q) -: p)

    proof

      let D be non empty set;

      let f be FinSequence of D;

      let p,q be Element of D;

      assume that

       A1: p in ( rng f) and

       A2: q in ( rng f) and

       A3: (q .. f) <= (p .. f);

      

       A4: (f -: p) = (f | (p .. f)) & ((f :- q) -: p) = ((f :- q) | (p .. (f :- q))) by FINSEQ_5:def 1;

      consider i be Element of NAT such that

       A5: (i + 1) = (q .. f) and

       A6: (f :- q) = (f /^ i) by A2, FINSEQ_5: 49;

      

       A7: i < (p .. f) by A3, A5, NAT_1: 13;

      then (p .. f) = (i + (p .. (f /^ i))) by A1, FINSEQ_6: 56;

      

      then

       A8: (p .. (f /^ i)) = ((p .. f) - i)

      .= ((p .. f) -' i) by A7, XREAL_1: 233;

      q in ( rng (f -: p)) by A1, A2, A3, FINSEQ_5: 46;

      then

       A9: ex j be Element of NAT st (j + 1) = (q .. (f -: p)) & ((f -: p) :- q) = ((f -: p) /^ j) by FINSEQ_5: 49;

      (q .. (f -: p)) = (q .. f) by A1, A2, A3, SPRECT_5: 3;

      hence thesis by A5, A6, A9, A4, A8, FINSEQ_5: 80;

    end;

    theorem :: JORDAN1G:17

    

     Th17: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for n be Nat holds (( L~ (( Cage (C,n)) -: ( W-min ( L~ ( Cage (C,n)))))) /\ ( L~ (( Cage (C,n)) :- ( W-min ( L~ ( Cage (C,n))))))) = {( N-min ( L~ ( Cage (C,n)))), ( W-min ( L~ ( Cage (C,n))))}

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let n be Nat;

      set US = (( Cage (C,n)) -: ( W-min ( L~ ( Cage (C,n)))));

      set LS = (( Cage (C,n)) :- ( W-min ( L~ ( Cage (C,n)))));

      set f = ( Cage (C,n));

      set pW = ( W-min ( L~ ( Cage (C,n))));

      set pN = ( N-min ( L~ ( Cage (C,n))));

      set pNa = ( N-max ( L~ ( Cage (C,n))));

      set pSa = ( S-max ( L~ ( Cage (C,n))));

      set pSi = ( S-min ( L~ ( Cage (C,n))));

      set pEa = ( E-max ( L~ ( Cage (C,n))));

      set pEi = ( E-min ( L~ ( Cage (C,n))));

      

       A1: ( W-min ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 43;

      then

       A2: (( Cage (C,n)) -: pW) <> {} by FINSEQ_5: 47;

      ( len (f -: pW)) = (pW .. f) by A1, FINSEQ_5: 42;

      then ((f -: pW) /. ( len (f -: pW))) = pW by A1, FINSEQ_5: 45;

      then

       A3: pW in ( rng (( Cage (C,n)) -: pW)) by A2, FINSEQ_6: 168;

      

       A4: (f /. 1) = pN by JORDAN9: 32;

      then (pEa .. f) < (pEi .. f) by SPRECT_2: 71;

      then (pNa .. f) < (pEi .. f) by A4, SPRECT_2: 70, XXREAL_0: 2;

      then (pNa .. f) < (pSa .. f) by A4, SPRECT_2: 72, XXREAL_0: 2;

      then

       A5: (pNa .. f) < (pSi .. f) by A4, SPRECT_2: 73, XXREAL_0: 2;

      ((( Cage (C,n)) -: pW) /. 1) = (( Cage (C,n)) /. 1) by A1, FINSEQ_5: 44

      .= pN by JORDAN9: 32;

      then

       A6: ( N-min ( L~ ( Cage (C,n)))) in ( rng (( Cage (C,n)) -: ( W-min ( L~ ( Cage (C,n)))))) by A2, FINSEQ_6: 42;

      ( N-max ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) & (pSi .. f) <= (pW .. f) by A4, SPRECT_2: 40, SPRECT_2: 74;

      then

       A7: pNa in ( rng (f -: pW)) by A1, A5, FINSEQ_5: 46, XXREAL_0: 2;

      

       A8: {pN, pNa, pW} c= ( rng US) by A6, A7, A3, ENUMSET1:def 1;

      then

       A9: ( card {pN, pNa, pW}) c= ( card ( rng US)) by CARD_1: 11;

      ((( Cage (C,n)) :- pW) /. 1) = pW by FINSEQ_5: 53;

      then

       A10: ( W-min ( L~ ( Cage (C,n)))) in ( rng (( Cage (C,n)) :- ( W-min ( L~ ( Cage (C,n)))))) by FINSEQ_6: 42;

      ((f :- pW) /. ( len (f :- pW))) = (f /. ( len f)) by A1, FINSEQ_5: 54

      .= (f /. 1) by FINSEQ_6:def 1

      .= pN by JORDAN9: 32;

      then

       A11: pN in ( rng (( Cage (C,n)) :- pW)) by FINSEQ_6: 168;

       {pN, pW} c= ( rng LS) by A11, A10, TARSKI:def 2;

      then

       A12: ( card {pN, pW}) c= ( card ( rng LS)) by CARD_1: 11;

      ( card ( rng LS)) c= ( card ( dom LS)) by CARD_2: 61;

      then

       A13: ( card ( rng LS)) c= ( len LS) by CARD_1: 62;

      ( W-max ( L~ f)) in ( L~ f) & (pN `2 ) = ( N-bound ( L~ f)) by EUCLID: 52, SPRECT_1: 13;

      then (( W-max ( L~ f)) `2 ) <= (pN `2 ) by PSCOMP_1: 24;

      then

       A14: pN <> pW by SPRECT_2: 57;

      then ( card {pN, pW}) = 2 by CARD_2: 57;

      then ( Segm 2) c= ( Segm ( len LS)) by A12, A13;

      then ( len LS) >= 2 by NAT_1: 39;

      then

       A15: ( rng LS) c= ( L~ LS) by SPPOL_2: 18;

      (LS /. ( len LS)) = (( Cage (C,n)) /. ( len ( Cage (C,n)))) by A1, FINSEQ_5: 54

      .= (( Cage (C,n)) /. 1) by FINSEQ_6:def 1

      .= ( N-min ( L~ ( Cage (C,n)))) by JORDAN9: 32;

      then

       A16: ( N-min ( L~ ( Cage (C,n)))) in ( rng LS) by FINSEQ_6: 168;

      (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) <= (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)));

      then

       A17: ( W-min ( L~ ( Cage (C,n)))) in ( rng LS) & ( W-min ( L~ ( Cage (C,n)))) in ( rng US) by A1, FINSEQ_5: 46, FINSEQ_6: 61;

      ( W-max ( L~ f)) in ( L~ f) & (pNa `2 ) = ( N-bound ( L~ f)) by EUCLID: 52, SPRECT_1: 13;

      then (( W-max ( L~ f)) `2 ) <= (pNa `2 ) by PSCOMP_1: 24;

      then pN <> pNa & pNa <> pW by SPRECT_2: 52, SPRECT_2: 57;

      then

       A18: ( card {pN, pNa, pW}) = 3 by A14, CARD_2: 58;

      ( card ( rng US)) c= ( card ( dom US)) by CARD_2: 61;

      then ( card ( rng US)) c= ( len US) by CARD_1: 62;

      then ( Segm 3) c= ( Segm ( len US)) by A18, A9;

      then

       A19: ( len US) >= 3 by NAT_1: 39;

      then

       A20: ( rng US) c= ( L~ US) by SPPOL_2: 18, XXREAL_0: 2;

      thus (( L~ US) /\ ( L~ LS)) c= {( N-min ( L~ ( Cage (C,n)))), ( W-min ( L~ ( Cage (C,n))))}

      proof

        let x be object;

        assume

         A21: x in (( L~ US) /\ ( L~ LS));

        then

        reconsider x1 = x as Point of ( TOP-REAL 2);

        assume

         A22: not x in {( N-min ( L~ ( Cage (C,n)))), ( W-min ( L~ ( Cage (C,n))))};

        x in ( L~ US) by A21, XBOOLE_0:def 4;

        then

        consider i1 be Nat such that

         A23: 1 <= i1 and

         A24: (i1 + 1) <= ( len US) and

         A25: x1 in ( LSeg (US,i1)) by SPPOL_2: 13;

        

         A26: ( LSeg (US,i1)) = ( LSeg (f,i1)) by A24, SPPOL_2: 9;

        x in ( L~ LS) by A21, XBOOLE_0:def 4;

        then

        consider i2 be Nat such that

         A27: 1 <= i2 and

         A28: (i2 + 1) <= ( len LS) and

         A29: x1 in ( LSeg (LS,i2)) by SPPOL_2: 13;

        set i3 = (i2 -' 1);

        

         A30: (i3 + 1) = i2 by A27, XREAL_1: 235;

        then

         A31: (1 + (pW .. f)) <= ((i3 + 1) + (pW .. f)) by A27, XREAL_1: 7;

        

         A32: ( len LS) = ((( len f) - (pW .. f)) + 1) by A1, FINSEQ_5: 50;

        then i2 < ((( len f) - (pW .. f)) + 1) by A28, NAT_1: 13;

        then (i2 - 1) < (( len f) - (pW .. f)) by XREAL_1: 19;

        then

         A33: ((i2 - 1) + (pW .. f)) < ( len f) by XREAL_1: 20;

        (i2 - 1) >= (1 - 1) by A27, XREAL_1: 9;

        then

         A34: (i3 + (pW .. f)) < ( len f) by A33, XREAL_0:def 2;

        

         A35: ( LSeg (LS,i2)) = ( LSeg (f,(i3 + (pW .. f)))) by A1, A30, SPPOL_2: 10;

        

         A36: ( len US) = (pW .. f) by A1, FINSEQ_5: 42;

        then (i1 + 1) < ((pW .. f) + 1) by A24, NAT_1: 13;

        then (i1 + 1) < ((i3 + (pW .. f)) + 1) by A31, XXREAL_0: 2;

        then

         A37: (i1 + 1) <= (i3 + (pW .. f)) by NAT_1: 13;

        

         A38: (((pW .. f) -' 1) + 1) = (pW .. f) by A1, FINSEQ_4: 21, XREAL_1: 235;

        (i3 + 1) < ((( len f) - (pW .. f)) + 1) by A28, A30, A32, NAT_1: 13;

        then i3 < (( len f) - (pW .. f)) by XREAL_1: 7;

        then

         A39: (i3 + (pW .. f)) < ( len f) by XREAL_1: 20;

        then

         A40: ((i3 + (pW .. f)) + 1) <= ( len f) by NAT_1: 13;

        now

          per cases by A23, A37, XXREAL_0: 1;

            suppose (i1 + 1) < (i3 + (pW .. f)) & i1 > 1;

            then ( LSeg (f,i1)) misses ( LSeg (f,(i3 + (pW .. f)))) by A39, GOBOARD5:def 4;

            then (( LSeg (f,i1)) /\ ( LSeg (f,(i3 + (pW .. f))))) = {} ;

            hence contradiction by A25, A29, A26, A35, XBOOLE_0:def 4;

          end;

            suppose

             A41: i1 = 1;

            (i3 + (pW .. f)) >= ( 0 + 3) by A19, A36, XREAL_1: 7;

            then

             A42: (i1 + 1) < (i3 + (pW .. f)) by A41, XXREAL_0: 2;

            now

              per cases by A40, XXREAL_0: 1;

                suppose ((i3 + (pW .. f)) + 1) < ( len f);

                then ( LSeg (f,i1)) misses ( LSeg (f,(i3 + (pW .. f)))) by A42, GOBOARD5:def 4;

                then (( LSeg (f,i1)) /\ ( LSeg (f,(i3 + (pW .. f))))) = {} ;

                hence contradiction by A25, A29, A26, A35, XBOOLE_0:def 4;

              end;

                suppose ((i3 + (pW .. f)) + 1) = ( len f);

                then (i3 + (pW .. f)) = (( len f) - 1);

                then (i3 + (pW .. f)) = (( len f) -' 1) by XREAL_0:def 2;

                then (( LSeg (f,i1)) /\ ( LSeg (f,(i3 + (pW .. f))))) = {(f /. 1)} by A41, GOBOARD7: 34, REVROT_1: 30;

                then x1 in {(f /. 1)} by A25, A29, A26, A35, XBOOLE_0:def 4;

                

                then x1 = (f /. 1) by TARSKI:def 1

                .= pN by JORDAN9: 32;

                hence contradiction by A22, TARSKI:def 2;

              end;

            end;

            hence contradiction;

          end;

            suppose

             A43: (i1 + 1) = (i3 + (pW .. f));

            (i3 + (pW .. f)) >= (pW .. f) by NAT_1: 11;

            then (pW .. f) < ( len f) by A34, XXREAL_0: 2;

            then ((pW .. f) + 1) <= ( len f) by NAT_1: 13;

            then

             A44: (((pW .. f) -' 1) + (1 + 1)) <= ( len f) by A38;

            ( 0 + (pW .. f)) <= (i3 + (pW .. f)) by XREAL_1: 7;

            then (pW .. f) = (i1 + 1) by A24, A36, A43, XXREAL_0: 1;

            then (( LSeg (f,i1)) /\ ( LSeg (f,(i3 + (pW .. f))))) = {(f /. (pW .. f))} by A23, A38, A43, A44, TOPREAL1:def 6;

            then x1 in {(f /. (pW .. f))} by A25, A29, A26, A35, XBOOLE_0:def 4;

            

            then x1 = (f /. (pW .. f)) by TARSKI:def 1

            .= pW by A1, FINSEQ_5: 38;

            hence contradiction by A22, TARSKI:def 2;

          end;

        end;

        hence contradiction;

      end;

      

       A45: (US /. 1) = (( Cage (C,n)) /. 1) by A1, FINSEQ_5: 44

      .= ( N-min ( L~ ( Cage (C,n)))) by JORDAN9: 32;

      US is non empty by A8;

      then

       A46: ( N-min ( L~ ( Cage (C,n)))) in ( rng US) by A45, FINSEQ_6: 42;

      thus {( N-min ( L~ ( Cage (C,n)))), ( W-min ( L~ ( Cage (C,n))))} c= (( L~ US) /\ ( L~ LS))

      proof

        let x be object;

        assume

         A47: x in {( N-min ( L~ ( Cage (C,n)))), ( W-min ( L~ ( Cage (C,n))))};

        per cases by A47, TARSKI:def 2;

          suppose x = ( N-min ( L~ ( Cage (C,n))));

          hence thesis by A15, A20, A46, A16, XBOOLE_0:def 4;

        end;

          suppose x = ( W-min ( L~ ( Cage (C,n))));

          hence thesis by A15, A20, A17, XBOOLE_0:def 4;

        end;

      end;

    end;

    theorem :: JORDAN1G:18

    

     Th18: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds ( Lower_Seq (C,n)) = (( Rotate (( Cage (C,n)),( E-max ( L~ ( Cage (C,n)))))) -: ( W-min ( L~ ( Cage (C,n)))))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      set Nmi = ( N-min ( L~ ( Cage (C,n))));

      set Nma = ( N-max ( L~ ( Cage (C,n))));

      set Wmi = ( W-min ( L~ ( Cage (C,n))));

      set Wma = ( W-max ( L~ ( Cage (C,n))));

      set Ema = ( E-max ( L~ ( Cage (C,n))));

      set Emi = ( E-min ( L~ ( Cage (C,n))));

      set Sma = ( S-max ( L~ ( Cage (C,n))));

      set Smi = ( S-min ( L~ ( Cage (C,n))));

      set RotWmi = ( Rotate (( Cage (C,n)),Wmi));

      set RotEma = ( Rotate (( Cage (C,n)),Ema));

      

       A1: Ema in ( rng ( Cage (C,n))) by SPRECT_2: 46;

      Wma in ( L~ ( Cage (C,n))) & (Nmi `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by EUCLID: 52, SPRECT_1: 13;

      then (Wma `2 ) <= (Nmi `2 ) by PSCOMP_1: 24;

      then Nmi <> Wmi by SPRECT_2: 57;

      then

       A2: ( card {Nmi, Wmi}) = 2 by CARD_2: 57;

      

       A3: Wmi in ( rng ( Cage (C,n))) by SPRECT_2: 43;

      then

       A4: (( Cage (C,n)) -: Wmi) <> {} by FINSEQ_5: 47;

      ( len (( Cage (C,n)) -: Wmi)) = (Wmi .. ( Cage (C,n))) by A3, FINSEQ_5: 42;

      then ((( Cage (C,n)) -: Wmi) /. ( len (( Cage (C,n)) -: Wmi))) = Wmi by A3, FINSEQ_5: 45;

      then

       A5: Wmi in ( rng (( Cage (C,n)) -: Wmi)) by A4, FINSEQ_6: 168;

      ((( Cage (C,n)) -: Wmi) /. 1) = (( Cage (C,n)) /. 1) by A3, FINSEQ_5: 44

      .= Nmi by JORDAN9: 32;

      then

       A6: Nmi in ( rng (( Cage (C,n)) -: Wmi)) by A4, FINSEQ_6: 42;

       {Nmi, Wmi} c= ( rng (( Cage (C,n)) -: Wmi)) by A6, A5, TARSKI:def 2;

      then

       A7: ( card {Nmi, Wmi}) c= ( card ( rng (( Cage (C,n)) -: Wmi))) by CARD_1: 11;

      ( card ( rng (( Cage (C,n)) -: Wmi))) c= ( card ( dom (( Cage (C,n)) -: Wmi))) by CARD_2: 61;

      then ( card ( rng (( Cage (C,n)) -: Wmi))) c= ( len (( Cage (C,n)) -: Wmi)) by CARD_1: 62;

      then ( Segm 2) c= ( Segm ( len (( Cage (C,n)) -: Wmi))) by A2, A7;

      then ( len (( Cage (C,n)) -: Wmi)) >= 2 by NAT_1: 39;

      then

       A8: ( rng (( Cage (C,n)) -: Wmi)) c= ( L~ (( Cage (C,n)) -: Wmi)) by SPPOL_2: 18;

      

       A9: (( Cage (C,n)) /. 1) = Nmi by JORDAN9: 32;

      then (Emi .. ( Cage (C,n))) <= (Sma .. ( Cage (C,n))) by SPRECT_2: 72;

      then (Ema .. ( Cage (C,n))) < (Sma .. ( Cage (C,n))) by A9, SPRECT_2: 71, XXREAL_0: 2;

      then

       A10: (Ema .. ( Cage (C,n))) < (Smi .. ( Cage (C,n))) by A9, SPRECT_2: 73, XXREAL_0: 2;

      then

       A11: (Ema .. ( Cage (C,n))) < (Wmi .. ( Cage (C,n))) by A9, SPRECT_2: 74, XXREAL_0: 2;

      

       A12: (Smi .. ( Cage (C,n))) <= (Wmi .. ( Cage (C,n))) by A9, SPRECT_2: 74;

      then

       A13: Ema in ( rng (( Cage (C,n)) -: Wmi)) by A3, A1, A10, FINSEQ_5: 46, XXREAL_0: 2;

      (Nma `1 ) <= (( NE-corner ( L~ ( Cage (C,n)))) `1 ) by PSCOMP_1: 38;

      then (Nmi `1 ) < (Nma `1 ) & (Nma `1 ) <= ( E-bound ( L~ ( Cage (C,n)))) by EUCLID: 52, SPRECT_2: 51;

      then

       A14: Nmi <> Ema by EUCLID: 52;

      

       A15: not Ema in ( rng (( Cage (C,n)) :- Wmi))

      proof

        ((( Cage (C,n)) :- Wmi) /. 1) = Wmi by FINSEQ_5: 53;

        then

         A16: Wmi in ( rng (( Cage (C,n)) :- Wmi)) by FINSEQ_6: 42;

        ((( Cage (C,n)) :- Wmi) /. ( len (( Cage (C,n)) :- Wmi))) = (( Cage (C,n)) /. ( len ( Cage (C,n)))) by A3, FINSEQ_5: 54

        .= (( Cage (C,n)) /. 1) by FINSEQ_6:def 1

        .= Nmi by JORDAN9: 32;

        then

         A17: Nmi in ( rng (( Cage (C,n)) :- Wmi)) by FINSEQ_6: 168;

         {Nmi, Wmi} c= ( rng (( Cage (C,n)) :- Wmi)) by A17, A16, TARSKI:def 2;

        then

         A18: ( card {Nmi, Wmi}) c= ( card ( rng (( Cage (C,n)) :- Wmi))) by CARD_1: 11;

        Wma in ( L~ ( Cage (C,n))) & (Nmi `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by EUCLID: 52, SPRECT_1: 13;

        then (Wma `2 ) <= (Nmi `2 ) by PSCOMP_1: 24;

        then Nmi <> Wmi by SPRECT_2: 57;

        then

         A19: ( card {Nmi, Wmi}) = 2 by CARD_2: 57;

        ( card ( rng (( Cage (C,n)) :- Wmi))) c= ( card ( dom (( Cage (C,n)) :- Wmi))) by CARD_2: 61;

        then ( card ( rng (( Cage (C,n)) :- Wmi))) c= ( len (( Cage (C,n)) :- Wmi)) by CARD_1: 62;

        then ( Segm 2) c= ( Segm ( len (( Cage (C,n)) :- Wmi))) by A19, A18;

        then ( len (( Cage (C,n)) :- Wmi)) >= 2 by NAT_1: 39;

        then

         A20: ( rng (( Cage (C,n)) :- Wmi)) c= ( L~ (( Cage (C,n)) :- Wmi)) by SPPOL_2: 18;

        assume Ema in ( rng (( Cage (C,n)) :- Wmi));

        then Ema in (( L~ (( Cage (C,n)) -: Wmi)) /\ ( L~ (( Cage (C,n)) :- Wmi))) by A13, A8, A20, XBOOLE_0:def 4;

        then Ema in {Nmi, Wmi} by Th17;

        then Ema = Wmi by A14, TARSKI:def 2;

        hence contradiction by TOPREAL5: 19;

      end;

      

       A21: (Nma .. ( Cage (C,n))) <= (Ema .. ( Cage (C,n))) by A9, SPRECT_2: 70;

      

       A22: (Nmi .. ( Cage (C,n))) < (Nma .. ( Cage (C,n))) by A9, SPRECT_2: 68;

      then

       A23: Nmi in ( rng ( Cage (C,n))) & (Nmi .. ( Cage (C,n))) < (Ema .. ( Cage (C,n))) by A9, SPRECT_2: 39, SPRECT_2: 70, XXREAL_0: 2;

      then

       A24: Nmi in ( rng (( Cage (C,n)) -: Wmi)) by A3, A11, FINSEQ_5: 46, XXREAL_0: 2;

      

       A25: (Ema .. (( Cage (C,n)) -: Wmi)) <> 1

      proof

        assume

         A26: (Ema .. (( Cage (C,n)) -: Wmi)) = 1;

        (Nmi .. (( Cage (C,n)) -: Wmi)) = (Nmi .. ( Cage (C,n))) by A3, A23, A11, SPRECT_5: 3, XXREAL_0: 2

        .= 1 by A9, FINSEQ_6: 43;

        hence contradiction by A22, A21, A13, A24, A26, FINSEQ_5: 9;

      end;

      then Ema in ( rng ((( Cage (C,n)) -: Wmi) /^ 1)) by A13, FINSEQ_6: 78;

      then

       A27: Ema in (( rng ((( Cage (C,n)) -: Wmi) /^ 1)) \ ( rng (( Cage (C,n)) :- Wmi))) by A15, XBOOLE_0:def 5;

      

       A28: Wmi in ( rng (( Cage (C,n)) :- Ema)) by A3, A1, A12, A10, FINSEQ_6: 62, XXREAL_0: 2;

      (RotWmi :- Ema) = (((( Cage (C,n)) :- Wmi) ^ ((( Cage (C,n)) -: Wmi) /^ 1)) :- Ema) by A3, FINSEQ_6:def 2

      .= (((( Cage (C,n)) -: Wmi) /^ 1) :- Ema) by A27, FINSEQ_6: 65

      .= ((( Cage (C,n)) -: Wmi) :- Ema) by A13, A25, FINSEQ_6: 83

      .= ((( Cage (C,n)) :- Ema) -: Wmi) by A3, A1, A12, A10, Th16, XXREAL_0: 2

      .= (((( Cage (C,n)) :- Ema) ^ ((( Cage (C,n)) -: Ema) /^ 1)) -: Wmi) by A28, FINSEQ_6: 66

      .= (RotEma -: Wmi) by A1, FINSEQ_6:def 2;

      hence thesis by JORDAN1E:def 2;

    end;

    theorem :: JORDAN1G:19

    

     Th19: for C be compact non vertical non horizontal Subset of ( TOP-REAL 2) holds (( W-min ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n))) = 1

    proof

      let C be compact non vertical non horizontal Subset of ( TOP-REAL 2);

      (( Upper_Seq (C,n)) /. 1) = ( W-min ( L~ ( Cage (C,n)))) by JORDAN1F: 5;

      hence thesis by FINSEQ_6: 43;

    end;

    theorem :: JORDAN1G:20

    

     Th20: for C be compact non vertical non horizontal Subset of ( TOP-REAL 2) holds (( W-min ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n))) < (( W-max ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n)))

    proof

      let C be compact non vertical non horizontal Subset of ( TOP-REAL 2);

      set Wmi = ( W-min ( L~ ( Cage (C,n))));

      set Wma = ( W-max ( L~ ( Cage (C,n))));

      set Nmi = ( N-min ( L~ ( Cage (C,n))));

      set Nma = ( N-max ( L~ ( Cage (C,n))));

      set Ema = ( E-max ( L~ ( Cage (C,n))));

      set Rot = ( Rotate (( Cage (C,n)),Wmi));

      

       A1: ( L~ Rot) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

      then

       A2: Wmi in ( rng Rot) by SPRECT_2: 43;

      

       A3: Wma in ( rng Rot) by A1, SPRECT_2: 44;

      

       A4: Ema in ( rng Rot) by A1, SPRECT_2: 46;

      Wmi in ( rng ( Cage (C,n))) by SPRECT_2: 43;

      then

       A5: (Rot /. 1) = Wmi by FINSEQ_6: 92;

      then

       A6: (Wmi .. Rot) < (Wma .. Rot) by A1, SPRECT_5: 21;

      

       A7: ( Upper_Seq (C,n)) = (Rot -: Ema) & (Nma .. Rot) <= (Ema .. Rot) by A1, A5, JORDAN1E:def 1, SPRECT_5: 25;

      (Nmi .. Rot) < (Nma .. Rot) by A1, A5, SPRECT_5: 24;

      then

       A8: (Wma .. Rot) < (Nma .. Rot) by A1, A5, SPRECT_5: 23, XXREAL_0: 2;

      then (Wma .. Rot) < (Ema .. Rot) by A1, A5, SPRECT_5: 25, XXREAL_0: 2;

      then (Wmi .. (Rot -: Ema)) = (Wmi .. Rot) by A2, A4, A6, SPRECT_5: 3, XXREAL_0: 2;

      hence thesis by A4, A6, A7, A8, A3, SPRECT_5: 3, XXREAL_0: 2;

    end;

    theorem :: JORDAN1G:21

    

     Th21: for C be compact non vertical non horizontal Subset of ( TOP-REAL 2) holds (( W-max ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n))) <= (( N-min ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n)))

    proof

      let C be compact non vertical non horizontal Subset of ( TOP-REAL 2);

      set Wmi = ( W-min ( L~ ( Cage (C,n))));

      set Wma = ( W-max ( L~ ( Cage (C,n))));

      set Nmi = ( N-min ( L~ ( Cage (C,n))));

      set Nma = ( N-max ( L~ ( Cage (C,n))));

      set Ema = ( E-max ( L~ ( Cage (C,n))));

      set Rot = ( Rotate (( Cage (C,n)),Wmi));

      

       A1: ( L~ Rot) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

      then

       A2: Wma in ( rng Rot) by SPRECT_2: 44;

      

       A3: Nmi in ( rng Rot) by A1, SPRECT_2: 39;

      

       A4: Ema in ( rng Rot) by A1, SPRECT_2: 46;

      Wmi in ( rng ( Cage (C,n))) by SPRECT_2: 43;

      then

       A5: (Rot /. 1) = Wmi by FINSEQ_6: 92;

      then

       A6: (Wma .. Rot) <= (Nmi .. Rot) by A1, SPRECT_5: 23;

      

       A7: ( Upper_Seq (C,n)) = (Rot -: Ema) & (Nma .. Rot) <= (Ema .. Rot) by A1, A5, JORDAN1E:def 1, SPRECT_5: 25;

      

       A8: (Nmi .. Rot) < (Nma .. Rot) by A1, A5, SPRECT_5: 24;

      then (Nmi .. Rot) < (Ema .. Rot) by A1, A5, SPRECT_5: 25, XXREAL_0: 2;

      then (Wma .. (Rot -: Ema)) = (Wma .. Rot) by A2, A4, A6, SPRECT_5: 3, XXREAL_0: 2;

      hence thesis by A4, A6, A8, A7, A3, SPRECT_5: 3, XXREAL_0: 2;

    end;

    theorem :: JORDAN1G:22

    

     Th22: for C be compact non vertical non horizontal Subset of ( TOP-REAL 2) holds (( N-min ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n))) < (( N-max ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n)))

    proof

      let C be compact non vertical non horizontal Subset of ( TOP-REAL 2);

      set Wmi = ( W-min ( L~ ( Cage (C,n))));

      set Nmi = ( N-min ( L~ ( Cage (C,n))));

      set Nma = ( N-max ( L~ ( Cage (C,n))));

      set Ema = ( E-max ( L~ ( Cage (C,n))));

      set Rot = ( Rotate (( Cage (C,n)),Wmi));

      

       A1: ( L~ Rot) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

      then

       A2: Ema in ( rng Rot) by SPRECT_2: 46;

      Wmi in ( rng ( Cage (C,n))) by SPRECT_2: 43;

      then (Rot /. 1) = Wmi by FINSEQ_6: 92;

      then

       A3: (Nmi .. Rot) < (Nma .. Rot) & (Nma .. Rot) <= (Ema .. Rot) by A1, SPRECT_5: 24, SPRECT_5: 25;

      

       A4: Nma in ( rng Rot) by A1, SPRECT_2: 40;

      Nmi in ( rng Rot) by A1, SPRECT_2: 39;

      then ( Upper_Seq (C,n)) = (Rot -: Ema) & (Nmi .. (Rot -: Ema)) = (Nmi .. Rot) by A2, A3, JORDAN1E:def 1, SPRECT_5: 3, XXREAL_0: 2;

      hence thesis by A2, A3, A4, SPRECT_5: 3;

    end;

    theorem :: JORDAN1G:23

    

     Th23: for C be compact non vertical non horizontal Subset of ( TOP-REAL 2) holds (( N-max ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n))) <= (( E-max ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n)))

    proof

      let C be compact non vertical non horizontal Subset of ( TOP-REAL 2);

      set Wmi = ( W-min ( L~ ( Cage (C,n))));

      set Nma = ( N-max ( L~ ( Cage (C,n))));

      set Ema = ( E-max ( L~ ( Cage (C,n))));

      set Rot = ( Rotate (( Cage (C,n)),Wmi));

      

       A1: ( L~ Rot) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

      then

       A2: Ema in ( rng Rot) by SPRECT_2: 46;

      Wmi in ( rng ( Cage (C,n))) by SPRECT_2: 43;

      then (Rot /. 1) = Wmi by FINSEQ_6: 92;

      then

       A3: (Nma .. Rot) <= (Ema .. Rot) by A1, SPRECT_5: 25;

      Nma in ( rng Rot) by A1, SPRECT_2: 40;

      then ( Upper_Seq (C,n)) = (Rot -: Ema) & (Nma .. (Rot -: Ema)) = (Nma .. Rot) by A2, A3, JORDAN1E:def 1, SPRECT_5: 3;

      hence thesis by A2, A3, SPRECT_5: 3;

    end;

    theorem :: JORDAN1G:24

    

     Th24: for C be compact non vertical non horizontal Subset of ( TOP-REAL 2) holds (( E-max ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n))) = ( len ( Upper_Seq (C,n)))

    proof

      let C be compact non vertical non horizontal Subset of ( TOP-REAL 2);

      ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 46;

      then

       A1: ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by FINSEQ_6: 90, SPRECT_2: 43;

      ( Upper_Seq (C,n)) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) -: ( E-max ( L~ ( Cage (C,n))))) & (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) <= (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by JORDAN1E:def 1;

      then ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Upper_Seq (C,n))) by A1, FINSEQ_5: 46;

      then

       A2: ( Upper_Seq (C,n)) just_once_values ( E-max ( L~ ( Cage (C,n)))) by FINSEQ_4: 8;

      (( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) = ( E-max ( L~ ( Cage (C,n)))) by JORDAN1F: 7;

      hence thesis by A2, FINSEQ_6: 166;

    end;

    theorem :: JORDAN1G:25

    

     Th25: for C be compact non vertical non horizontal Subset of ( TOP-REAL 2) holds (( E-max ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n))) = 1

    proof

      let C be compact non vertical non horizontal Subset of ( TOP-REAL 2);

      (( Lower_Seq (C,n)) /. 1) = ( E-max ( L~ ( Cage (C,n)))) by JORDAN1F: 6;

      hence thesis by FINSEQ_6: 43;

    end;

    theorem :: JORDAN1G:26

    

     Th26: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds (( E-max ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n))) < (( E-min ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n)))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      set Ema = ( E-max ( L~ ( Cage (C,n))));

      set Emi = ( E-min ( L~ ( Cage (C,n))));

      set Sma = ( S-max ( L~ ( Cage (C,n))));

      set Smi = ( S-min ( L~ ( Cage (C,n))));

      set Wmi = ( W-min ( L~ ( Cage (C,n))));

      set Rot = ( Rotate (( Cage (C,n)),Ema));

      

       A1: ( Lower_Seq (C,n)) = (Rot -: Wmi) by Th18;

      

       A2: ( L~ Rot) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

      then

       A3: Ema in ( rng Rot) by SPRECT_2: 46;

      

       A4: Emi in ( rng Rot) by A2, SPRECT_2: 45;

      

       A5: Wmi in ( rng Rot) by A2, SPRECT_2: 43;

      Ema in ( rng ( Cage (C,n))) by SPRECT_2: 46;

      then

       A6: (Rot /. 1) = Ema by FINSEQ_6: 92;

      then

       A7: (Ema .. Rot) < (Emi .. Rot) by A2, SPRECT_5: 37;

      

       A8: (Smi .. Rot) <= (Wmi .. Rot) by A2, A6, SPRECT_5: 41;

      (Sma .. Rot) < (Smi .. Rot) by A2, A6, SPRECT_5: 40;

      then

       A9: (Emi .. Rot) < (Smi .. Rot) by A2, A6, SPRECT_5: 39, XXREAL_0: 2;

      then (Emi .. Rot) < (Wmi .. Rot) by A2, A6, SPRECT_5: 41, XXREAL_0: 2;

      then (Ema .. (Rot -: Wmi)) = (Ema .. Rot) by A3, A5, A7, SPRECT_5: 3, XXREAL_0: 2;

      hence thesis by A1, A5, A7, A8, A9, A4, SPRECT_5: 3, XXREAL_0: 2;

    end;

    theorem :: JORDAN1G:27

    

     Th27: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds (( E-min ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n))) <= (( S-max ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n)))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      set Ema = ( E-max ( L~ ( Cage (C,n))));

      set Emi = ( E-min ( L~ ( Cage (C,n))));

      set Sma = ( S-max ( L~ ( Cage (C,n))));

      set Smi = ( S-min ( L~ ( Cage (C,n))));

      set Wmi = ( W-min ( L~ ( Cage (C,n))));

      set Rot = ( Rotate (( Cage (C,n)),Ema));

      

       A1: ( Lower_Seq (C,n)) = (Rot -: Wmi) by Th18;

      

       A2: ( L~ Rot) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

      then

       A3: Emi in ( rng Rot) by SPRECT_2: 45;

      

       A4: Sma in ( rng Rot) by A2, SPRECT_2: 42;

      

       A5: Wmi in ( rng Rot) by A2, SPRECT_2: 43;

      Ema in ( rng ( Cage (C,n))) by SPRECT_2: 46;

      then

       A6: (Rot /. 1) = Ema by FINSEQ_6: 92;

      then

       A7: (Emi .. Rot) <= (Sma .. Rot) by A2, SPRECT_5: 39;

      

       A8: (Smi .. Rot) <= (Wmi .. Rot) by A2, A6, SPRECT_5: 41;

      

       A9: (Sma .. Rot) < (Smi .. Rot) by A2, A6, SPRECT_5: 40;

      then (Sma .. Rot) < (Wmi .. Rot) by A2, A6, SPRECT_5: 41, XXREAL_0: 2;

      then (Emi .. (Rot -: Wmi)) = (Emi .. Rot) by A3, A5, A7, SPRECT_5: 3, XXREAL_0: 2;

      hence thesis by A1, A5, A7, A9, A8, A4, SPRECT_5: 3, XXREAL_0: 2;

    end;

    theorem :: JORDAN1G:28

    

     Th28: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds (( S-max ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n))) < (( S-min ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n)))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      set Ema = ( E-max ( L~ ( Cage (C,n))));

      set Sma = ( S-max ( L~ ( Cage (C,n))));

      set Smi = ( S-min ( L~ ( Cage (C,n))));

      set Wmi = ( W-min ( L~ ( Cage (C,n))));

      set Rot = ( Rotate (( Cage (C,n)),Ema));

      

       A1: ( Lower_Seq (C,n)) = (Rot -: Wmi) by Th18;

      

       A2: ( L~ Rot) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

      then

       A3: Wmi in ( rng Rot) by SPRECT_2: 43;

      Ema in ( rng ( Cage (C,n))) by SPRECT_2: 46;

      then (Rot /. 1) = Ema by FINSEQ_6: 92;

      then

       A4: (Sma .. Rot) < (Smi .. Rot) & (Smi .. Rot) <= (Wmi .. Rot) by A2, SPRECT_5: 40, SPRECT_5: 41;

      

       A5: Smi in ( rng Rot) by A2, SPRECT_2: 41;

      Sma in ( rng Rot) by A2, SPRECT_2: 42;

      then (Sma .. (Rot -: Wmi)) = (Sma .. Rot) by A3, A4, SPRECT_5: 3, XXREAL_0: 2;

      hence thesis by A1, A3, A4, A5, SPRECT_5: 3;

    end;

    theorem :: JORDAN1G:29

    

     Th29: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds (( S-min ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n))) <= (( W-min ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n)))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      set Ema = ( E-max ( L~ ( Cage (C,n))));

      set Smi = ( S-min ( L~ ( Cage (C,n))));

      set Wmi = ( W-min ( L~ ( Cage (C,n))));

      set Rot = ( Rotate (( Cage (C,n)),Ema));

      

       A1: ( Lower_Seq (C,n)) = (Rot -: Wmi) by Th18;

      

       A2: ( L~ Rot) = ( L~ ( Cage (C,n))) by REVROT_1: 33;

      then

       A3: Wmi in ( rng Rot) by SPRECT_2: 43;

      Ema in ( rng ( Cage (C,n))) by SPRECT_2: 46;

      then (Rot /. 1) = Ema by FINSEQ_6: 92;

      then

       A4: (Smi .. Rot) <= (Wmi .. Rot) by A2, SPRECT_5: 41;

      Smi in ( rng Rot) by A2, SPRECT_2: 41;

      then (Smi .. (Rot -: Wmi)) = (Smi .. Rot) by A3, A4, SPRECT_5: 3;

      hence thesis by A1, A3, A4, SPRECT_5: 3;

    end;

    theorem :: JORDAN1G:30

    

     Th30: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds (( W-min ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n))) = ( len ( Lower_Seq (C,n)))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      ( W-min ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 43;

      then

       A1: ( W-min ( L~ ( Cage (C,n)))) in ( rng ( Rotate (( Cage (C,n)),( E-max ( L~ ( Cage (C,n))))))) by FINSEQ_6: 90, SPRECT_2: 46;

      ( Lower_Seq (C,n)) = (( Rotate (( Cage (C,n)),( E-max ( L~ ( Cage (C,n)))))) -: ( W-min ( L~ ( Cage (C,n))))) & (( W-min ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( E-max ( L~ ( Cage (C,n))))))) <= (( W-min ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( E-max ( L~ ( Cage (C,n))))))) by Th18;

      then ( W-min ( L~ ( Cage (C,n)))) in ( rng ( Lower_Seq (C,n))) by A1, FINSEQ_5: 46;

      then

       A2: ( Lower_Seq (C,n)) just_once_values ( W-min ( L~ ( Cage (C,n)))) by FINSEQ_4: 8;

      (( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) = ( W-min ( L~ ( Cage (C,n)))) by JORDAN1F: 8;

      hence thesis by A2, FINSEQ_6: 166;

    end;

    theorem :: JORDAN1G:31

    

     Th31: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds ((( Upper_Seq (C,n)) /. 2) `1 ) = ( W-bound ( L~ ( Cage (C,n))))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      set Ca = ( Cage (C,n));

      set US = ( Upper_Seq (C,n));

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      set Nmin = ( N-min ( L~ ( Cage (C,n))));

      Emax in ( rng Ca) by SPRECT_2: 46;

      then

       A1: Emax in ( rng ( Rotate (Ca,Wmin))) by FINSEQ_6: 90, SPRECT_2: 43;

      ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 2 by XXREAL_0: 2;

      then 2 in ( Seg ( len US)) by FINSEQ_1: 1;

      then

       A2: 2 in ( Seg (Emax .. ( Rotate (Ca,Wmin)))) by JORDAN1E: 8;

      ((Ca :- Wmin) /. 1) = Wmin by FINSEQ_5: 53;

      then

       A3: Wmin in ( rng (Ca :- Wmin)) by FINSEQ_6: 42;

      (Ca /. 1) = Nmin by JORDAN9: 32;

      then (Wmin .. Ca) < ( len Ca) by SPRECT_2: 76;

      then

       A4: ((Wmin .. Ca) + 1) <= ( len Ca) by NAT_1: 13;

      ( W-max ( L~ Ca)) in ( L~ Ca) & (Nmin `2 ) = ( N-bound ( L~ Ca)) by EUCLID: 52, SPRECT_1: 13;

      then (( W-max ( L~ Ca)) `2 ) <= (Nmin `2 ) by PSCOMP_1: 24;

      then Nmin <> Wmin by SPRECT_2: 57;

      then

       A5: ( card {Nmin, Wmin}) = 2 by CARD_2: 57;

      

       A6: Wmin in ( rng Ca) by SPRECT_2: 43;

      then

       A7: 1 <= (Wmin .. Ca) by FINSEQ_4: 21;

      ((Ca :- Wmin) /. ( len (Ca :- Wmin))) = (Ca /. ( len Ca)) by A6, FINSEQ_5: 54

      .= (Ca /. 1) by FINSEQ_6:def 1

      .= Nmin by JORDAN9: 32;

      then

       A8: Nmin in ( rng (Ca :- Wmin)) by FINSEQ_6: 168;

       {Nmin, Wmin} c= ( rng (Ca :- Wmin)) by A8, A3, TARSKI:def 2;

      then

       A9: ( card {Nmin, Wmin}) c= ( card ( rng (Ca :- Wmin))) by CARD_1: 11;

      ( card ( rng (Ca :- Wmin))) c= ( card ( dom (Ca :- Wmin))) by CARD_2: 61;

      then ( card ( rng (Ca :- Wmin))) c= ( len (Ca :- Wmin)) by CARD_1: 62;

      then ( Segm 2) c= ( Segm ( len (Ca :- Wmin))) by A5, A9;

      then

       A10: ( len (Ca :- Wmin)) >= 2 by NAT_1: 39;

      then

       A11: ( len (Ca :- Wmin)) >= 1 by XXREAL_0: 2;

      

       A12: (US /. 1) = ((( Rotate (Ca,Wmin)) -: Emax) /. 1) by JORDAN1E:def 1

      .= (( Rotate (Ca,Wmin)) /. 1) by A1, FINSEQ_5: 44

      .= (Ca /. ((1 -' 1) + (Wmin .. Ca))) by A6, A11, FINSEQ_6: 174

      .= (Ca /. ( 0 + (Wmin .. Ca))) by XREAL_1: 232;

      (US /. 2) = ((( Rotate (Ca,Wmin)) -: Emax) /. 2) by JORDAN1E:def 1

      .= (( Rotate (Ca,Wmin)) /. 2) by A1, A2, FINSEQ_5: 43

      .= (Ca /. ((2 -' 1) + (Wmin .. Ca))) by A6, A10, FINSEQ_6: 174

      .= (Ca /. ((2 - 1) + (Wmin .. Ca))) by XREAL_0:def 2;

      hence thesis by A7, A4, A12, JORDAN1E: 22, JORDAN1F: 5;

    end;

    theorem :: JORDAN1G:32

    for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds ((( Lower_Seq (C,n)) /. 2) `1 ) = ( E-bound ( L~ ( Cage (C,n))))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      set Ca = ( Cage (C,n));

      set LS = ( Lower_Seq (C,n));

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      set Emin = ( E-min ( L~ ( Cage (C,n))));

      set Smax = ( S-max ( L~ ( Cage (C,n))));

      set Smin = ( S-min ( L~ ( Cage (C,n))));

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      set Nmin = ( N-min ( L~ ( Cage (C,n))));

      Wmin in ( rng Ca) by SPRECT_2: 43;

      then

       A1: Wmin in ( rng ( Rotate (Ca,Emax))) by FINSEQ_6: 90, SPRECT_2: 46;

      ( len LS) >= 3 by JORDAN1E: 15;

      then ( len LS) >= 2 by XXREAL_0: 2;

      then 2 <= (Wmin .. LS) by Th30;

      then 2 <= (Wmin .. (( Rotate (Ca,Emax)) -: Wmin)) by Th18;

      then 2 <= (Wmin .. ( Rotate (Ca,Emax))) by A1, FINSEQ_6: 72;

      then

       A2: 2 in ( Seg (Wmin .. ( Rotate (Ca,Emax)))) by FINSEQ_1: 1;

      ((Ca :- Emax) /. 1) = Emax by FINSEQ_5: 53;

      then

       A3: Emax in ( rng (Ca :- Emax)) by FINSEQ_6: 42;

      ( N-max ( L~ Ca)) in ( L~ Ca) & (Emax `1 ) = ( E-bound ( L~ Ca)) by EUCLID: 52, SPRECT_1: 11;

      then (( N-max ( L~ Ca)) `1 ) <= (Emax `1 ) by PSCOMP_1: 24;

      then Nmin <> Emax by SPRECT_2: 51;

      then

       A4: ( card {Nmin, Emax}) = 2 by CARD_2: 57;

      

       A5: (Ca /. 1) = Nmin by JORDAN9: 32;

      then (Emax .. Ca) < (Emin .. Ca) by SPRECT_2: 71;

      then (Emax .. Ca) < (Smax .. Ca) by A5, SPRECT_2: 72, XXREAL_0: 2;

      then (Emax .. Ca) < (Smin .. Ca) by A5, SPRECT_2: 73, XXREAL_0: 2;

      then (Emax .. Ca) < (Wmin .. Ca) by A5, SPRECT_2: 74, XXREAL_0: 2;

      then (Emax .. Ca) < ( len Ca) by A5, SPRECT_2: 76, XXREAL_0: 2;

      then

       A6: ((Emax .. Ca) + 1) <= ( len Ca) by NAT_1: 13;

      

       A7: Emax in ( rng Ca) by SPRECT_2: 46;

      then

       A8: 1 <= (Emax .. Ca) by FINSEQ_4: 21;

      ((Ca :- Emax) /. ( len (Ca :- Emax))) = (Ca /. ( len Ca)) by A7, FINSEQ_5: 54

      .= (Ca /. 1) by FINSEQ_6:def 1

      .= Nmin by JORDAN9: 32;

      then

       A9: Nmin in ( rng (Ca :- Emax)) by FINSEQ_6: 168;

       {Nmin, Emax} c= ( rng (Ca :- Emax)) by A9, A3, TARSKI:def 2;

      then

       A10: ( card {Nmin, Emax}) c= ( card ( rng (Ca :- Emax))) by CARD_1: 11;

      ( card ( rng (Ca :- Emax))) c= ( card ( dom (Ca :- Emax))) by CARD_2: 61;

      then ( card ( rng (Ca :- Emax))) c= ( len (Ca :- Emax)) by CARD_1: 62;

      then ( Segm 2) c= ( Segm ( len (Ca :- Emax))) by A4, A10;

      then

       A11: ( len (Ca :- Emax)) >= 2 by NAT_1: 39;

      then

       A12: ( len (Ca :- Emax)) >= 1 by XXREAL_0: 2;

      

       A13: (LS /. 1) = ((( Rotate (Ca,Emax)) -: Wmin) /. 1) by Th18

      .= (( Rotate (Ca,Emax)) /. 1) by A1, FINSEQ_5: 44

      .= (Ca /. ((1 -' 1) + (Emax .. Ca))) by A7, A12, FINSEQ_6: 174

      .= (Ca /. ( 0 + (Emax .. Ca))) by XREAL_1: 232;

      (LS /. 2) = ((( Rotate (Ca,Emax)) -: Wmin) /. 2) by Th18

      .= (( Rotate (Ca,Emax)) /. 2) by A1, A2, FINSEQ_5: 43

      .= (Ca /. ((2 -' 1) + (Emax .. Ca))) by A7, A11, FINSEQ_6: 174

      .= (Ca /. ((2 - 1) + (Emax .. Ca))) by XREAL_0:def 2;

      hence thesis by A8, A6, A13, JORDAN1E: 20, JORDAN1F: 6;

    end;

    theorem :: JORDAN1G:33

    

     Th33: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds (( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) = (( W-bound C) + ( E-bound C))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      

      thus (( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) = (( W-bound ( L~ ( Cage (C,n)))) + (( E-bound C) + ((( E-bound C) - ( W-bound C)) / (2 |^ n)))) by JORDAN1A: 64

      .= ((( W-bound C) - ((( E-bound C) - ( W-bound C)) / (2 |^ n))) + (( E-bound C) + ((( E-bound C) - ( W-bound C)) / (2 |^ n)))) by JORDAN1A: 62

      .= (( W-bound C) + ( E-bound C));

    end;

    theorem :: JORDAN1G:34

    for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds (( S-bound ( L~ ( Cage (C,n)))) + ( N-bound ( L~ ( Cage (C,n))))) = (( S-bound C) + ( N-bound C))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      

      thus (( S-bound ( L~ ( Cage (C,n)))) + ( N-bound ( L~ ( Cage (C,n))))) = (( S-bound ( L~ ( Cage (C,n)))) + (( N-bound C) + ((( N-bound C) - ( S-bound C)) / (2 |^ n)))) by JORDAN10: 6

      .= ((( S-bound C) - ((( N-bound C) - ( S-bound C)) / (2 |^ n))) + (( N-bound C) + ((( N-bound C) - ( S-bound C)) / (2 |^ n)))) by JORDAN1A: 63

      .= (( S-bound C) + ( N-bound C));

    end;

    theorem :: JORDAN1G:35

    

     Th35: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for n be Nat, i be Nat st 1 <= i & i <= ( width ( Gauge (C,n))) & n > 0 holds ((( Gauge (C,n)) * (( Center ( Gauge (C,n))),i)) `1 ) = ((( W-bound C) + ( E-bound C)) / 2)

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let n be Nat, i be Nat such that

       A1: 1 <= i & i <= ( width ( Gauge (C,n)));

      reconsider ii = i as Nat;

      

       A2: ( len ( Gauge (C,n))) = ( width ( Gauge (C,n))) by JORDAN8:def 1;

      assume

       A3: n > 0 ;

      ( len ( Gauge (C,1))) >= 4 by JORDAN8: 10;

      then

       A4: ( len ( Gauge (C,1))) >= 1 by XXREAL_0: 2;

      

      thus ((( Gauge (C,n)) * (( Center ( Gauge (C,n))),i)) `1 ) = ((( Gauge (C,n)) * (( Center ( Gauge (C,n))),ii)) `1 )

      .= ((( Gauge (C,1)) * (( Center ( Gauge (C,1))),1)) `1 ) by A1, A2, A4, A3, JORDAN1A: 36

      .= ((( W-bound C) + ( E-bound C)) / 2) by A4, JORDAN1A: 38;

    end;

    theorem :: JORDAN1G:36

    for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for n,i be Nat st 1 <= i & i <= ( len ( Gauge (C,n))) & n > 0 holds ((( Gauge (C,n)) * (i,( Center ( Gauge (C,n))))) `2 ) = ((( S-bound C) + ( N-bound C)) / 2)

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let n,i be Nat such that

       A1: 1 <= i & i <= ( len ( Gauge (C,n)));

      ( len ( Gauge (C,1))) >= 4 by JORDAN8: 10;

      then

       A2: ( len ( Gauge (C,1))) >= 1 by XXREAL_0: 2;

      assume n > 0 ;

      

      hence ((( Gauge (C,n)) * (i,( Center ( Gauge (C,n))))) `2 ) = ((( Gauge (C,1)) * (1,( Center ( Gauge (C,1))))) `2 ) by A1, A2, JORDAN1A: 37

      .= ((( S-bound C) + ( N-bound C)) / 2) by A2, JORDAN1A: 39;

    end;

    theorem :: JORDAN1G:37

    

     Th37: for f be S-Sequence_in_R2 holds for k1,k2 be Nat st 1 <= k1 & k1 <= ( len f) & 1 <= k2 & k2 <= ( len f) & (f /. 1) in ( L~ ( mid (f,k1,k2))) holds k1 = 1 or k2 = 1

    proof

      let f be S-Sequence_in_R2;

      let k1,k2 be Nat;

      assume that

       A1: 1 <= k1 and

       A2: k1 <= ( len f) and

       A3: 1 <= k2 and

       A4: k2 <= ( len f) and

       A5: (f /. 1) in ( L~ ( mid (f,k1,k2)));

      

       AA: k1 in ( dom f) by FINSEQ_3: 25, A1, A2;

      assume that

       A6: k1 <> 1 and

       A7: k2 <> 1;

      

       A8: ( len f) >= 2 by TOPREAL1:def 8;

      consider j be Nat such that

       A9: 1 <= j and

       A10: (j + 1) <= ( len ( mid (f,k1,k2))) and

       A11: (f /. 1) in ( LSeg (( mid (f,k1,k2)),j)) by A5, SPPOL_2: 13;

      per cases by XXREAL_0: 1;

        suppose

         A12: k1 < k2;

        then ( len ( mid (f,k1,k2))) = ((k2 -' k1) + 1) by A1, A2, A3, A4, FINSEQ_6: 118;

        then j < ((k2 -' k1) + 1) by A10, NAT_1: 13;

        then ( LSeg (( mid (f,k1,k2)),j)) = ( LSeg (f,((j + k1) -' 1))) by A1, A4, A9, A12, JORDAN4: 19;

        then

         A13: ((j + k1) -' 1) = 1 by A11, A8, JORDAN5B: 30;

        (j + k1) >= (1 + 1) by A1, A9, XREAL_1: 7;

        then ((j + k1) - 1) >= ((1 + 1) - 1) by XREAL_1: 9;

        then (j + (k1 - 1)) = 1 by A13, XREAL_0:def 2;

        then (k1 - 1) = (1 - j);

        then (k1 - 1) <= 0 by A9, XREAL_1: 47;

        then (k1 - 1) = 0 by A1, XREAL_1: 48;

        hence contradiction by A6;

      end;

        suppose

         A14: k1 > k2;

        then ( len ( mid (f,k1,k2))) = ((k1 -' k2) + 1) by A1, A2, A3, A4, FINSEQ_6: 118;

        then

         A15: j < ((k1 -' k2) + 1) by A10, NAT_1: 13;

        (k1 - k2) > 0 by A14, XREAL_1: 50;

        then (k1 -' k2) = (k1 - k2) by XREAL_0:def 2;

        then (j - 1) < (k1 - k2) by A15, XREAL_1: 19;

        then ((j - 1) + k2) < k1 by XREAL_1: 20;

        then (j + ( - (1 - k2))) < k1;

        then

         A16: (k2 - 1) < (k1 - j) by XREAL_1: 20;

        ( LSeg (( mid (f,k1,k2)),j)) = ( LSeg (f,(k1 -' j))) by A2, A3, A9, A14, A15, JORDAN4: 20;

        then (k1 -' j) = 1 by A11, A8, JORDAN5B: 30;

        then (k1 - j) = 1 by XREAL_0:def 2;

        then k2 < (1 + 1) by A16, XREAL_1: 19;

        then k2 <= 1 by NAT_1: 13;

        hence contradiction by A3, A7, XXREAL_0: 1;

      end;

        suppose k1 = k2;

        

        then ( mid (f,k1,k2)) = <*(f . k1)*> by AA, JORDAN4: 15

        .= <*(f /. k1)*> by AA, PARTFUN1:def 6;

        hence contradiction by A5, SPPOL_2: 12;

      end;

    end;

    theorem :: JORDAN1G:38

    

     Th38: for f be S-Sequence_in_R2 holds for k1,k2 be Nat st 1 <= k1 & k1 <= ( len f) & 1 <= k2 & k2 <= ( len f) & (f /. ( len f)) in ( L~ ( mid (f,k1,k2))) holds k1 = ( len f) or k2 = ( len f)

    proof

      let f be S-Sequence_in_R2;

      let k1,k2 be Nat;

      assume that

       A1: 1 <= k1 and

       A2: k1 <= ( len f) and

       A3: 1 <= k2 and

       A4: k2 <= ( len f) and

       A5: (f /. ( len f)) in ( L~ ( mid (f,k1,k2)));

      

       AA: k1 in ( dom f) by A1, A2, FINSEQ_3: 25;

      assume that

       A6: k1 <> ( len f) and

       A7: k2 <> ( len f);

      consider j be Nat such that

       A8: 1 <= j and

       A9: (j + 1) <= ( len ( mid (f,k1,k2))) and

       A10: (f /. ( len f)) in ( LSeg (( mid (f,k1,k2)),j)) by A5, SPPOL_2: 13;

      per cases by XXREAL_0: 1;

        suppose

         A11: k1 < k2;

        then

         A12: ( len ( mid (f,k1,k2))) = ((k2 -' k1) + 1) by A1, A2, A3, A4, FINSEQ_6: 118;

        then

         A13: j < ((k2 -' k1) + 1) by A9, NAT_1: 13;

        

         A14: (j + k1) >= (1 + 1) by A1, A8, XREAL_1: 7;

        then

         A15: ((j + k1) - 1) >= ((1 + 1) - 1) by XREAL_1: 9;

        then

         A16: ((j + k1) -' 1) = ((j + k1) - 1) by XREAL_0:def 2;

        (k2 - k1) > 0 by A11, XREAL_1: 50;

        then

         A17: (k2 -' k1) = (k2 - k1) by XREAL_0:def 2;

        then (j - 1) < (k2 - k1) by A13, XREAL_1: 19;

        then ((j - 1) + k1) < k2 by XREAL_1: 20;

        then

         A18: ((j + k1) - 1) < ( len f) by A4, XXREAL_0: 2;

        then

         A19: ((j + k1) -' 1) in ( dom f) by A15, A16, FINSEQ_3: 25;

        

         A20: (j + k1) >= 1 by A14, XXREAL_0: 2;

        (((j + k1) - 1) + 1) <= ( len f) by A16, A18, NAT_1: 13;

        then (j + k1) in ( Seg ( len f)) by A20, FINSEQ_1: 1;

        then

         A21: (((j + k1) -' 1) + 1) in ( dom f) by A16, FINSEQ_1:def 3;

        ( LSeg (( mid (f,k1,k2)),j)) = ( LSeg (f,((j + k1) -' 1))) by A1, A4, A8, A11, A13, JORDAN4: 19;

        then

         A22: (((j + k1) -' 1) + 1) = ( len f) by A10, A19, A21, GOBOARD2: 2;

        

         A23: ((j + k1) -' 1) = ((j + k1) - 1) by A15, XREAL_0:def 2;

        j < ((k2 + 1) - k1) by A9, A17, A12, NAT_1: 13;

        then ( len f) < (k2 + 1) by A22, A23, XREAL_1: 20;

        then ( len f) <= k2 by NAT_1: 13;

        hence contradiction by A4, A7, XXREAL_0: 1;

      end;

        suppose

         A24: k1 > k2;

        then ( len ( mid (f,k1,k2))) = ((k1 -' k2) + 1) by A1, A2, A3, A4, FINSEQ_6: 118;

        then

         A25: j < ((k1 -' k2) + 1) by A9, NAT_1: 13;

        (k1 - k2) > 0 by A24, XREAL_1: 50;

        then (k1 -' k2) = (k1 - k2) by XREAL_0:def 2;

        then (j - 1) < (k1 - k2) by A25, XREAL_1: 19;

        then ((j - 1) + k2) < k1 by XREAL_1: 20;

        then

         A26: (j + ( - (1 - k2))) < k1;

        then

         A27: ( - (1 - k2)) < (k1 - j) by XREAL_1: 20;

        

         A28: (k2 - 1) >= 0 by A3, XREAL_1: 48;

        then

         A29: ((k1 - j) + 1) > ( 0 + 1) by A27, XREAL_1: 6;

        (k2 - 1) < (k1 - j) by A26, XREAL_1: 20;

        then

         A30: (k1 - j) > 0 by A3, XREAL_1: 48;

        then

         A31: (k1 -' j) = (k1 - j) by XREAL_0:def 2;

        (k1 - j) <= (k1 - 1) by A8, XREAL_1: 10;

        then ((k1 - j) + 1) <= ((k1 - 1) + 1) by XREAL_1: 7;

        then (k1 - j) < k1 by A31, NAT_1: 13;

        then

         A32: (k1 - j) < ( len f) by A2, XXREAL_0: 2;

        then ((k1 - j) + 1) <= ( len f) by A31, NAT_1: 13;

        then

         A33: ((k1 -' j) + 1) in ( dom f) by A31, A29, FINSEQ_3: 25;

        (k1 - j) >= ( 0 + 1) by A27, A28, A31, NAT_1: 13;

        then

         A34: (k1 -' j) in ( dom f) by A31, A32, FINSEQ_3: 25;

        ( LSeg (( mid (f,k1,k2)),j)) = ( LSeg (f,(k1 -' j))) by A2, A3, A8, A24, A25, JORDAN4: 20;

        then ((k1 -' j) + 1) = ( len f) by A10, A34, A33, GOBOARD2: 2;

        then

         A35: ((k1 - j) + 1) = ( len f) by A30, XREAL_0:def 2;

        (k1 - j) <= (k1 - 1) by A8, XREAL_1: 10;

        then ( len f) <= ((k1 - 1) + 1) by A35, XREAL_1: 7;

        hence contradiction by A2, A6, XXREAL_0: 1;

      end;

        suppose k1 = k2;

        

        then ( mid (f,k1,k2)) = <*(f . k1)*> by AA, JORDAN4: 15

        .= <*(f /. k1)*> by AA, PARTFUN1:def 6;

        hence contradiction by A5, SPPOL_2: 12;

      end;

    end;

    theorem :: JORDAN1G:39

    

     Th39: for C be compact non vertical non horizontal Subset of ( TOP-REAL 2) holds for n be Nat holds ( rng ( Upper_Seq (C,n))) c= ( rng ( Cage (C,n))) & ( rng ( Lower_Seq (C,n))) c= ( rng ( Cage (C,n)))

    proof

      let C be compact non vertical non horizontal Subset of ( TOP-REAL 2);

      let n be Nat;

      ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 46;

      then

       A1: ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by FINSEQ_6: 90, SPRECT_2: 43;

      ( Upper_Seq (C,n)) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) -: ( E-max ( L~ ( Cage (C,n))))) by JORDAN1E:def 1;

      then ( rng ( Upper_Seq (C,n))) c= ( rng ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by FINSEQ_5: 48;

      hence ( rng ( Upper_Seq (C,n))) c= ( rng ( Cage (C,n))) by FINSEQ_6: 90, SPRECT_2: 43;

      ( Lower_Seq (C,n)) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) :- ( E-max ( L~ ( Cage (C,n))))) by JORDAN1E:def 2;

      then ( rng ( Lower_Seq (C,n))) c= ( rng ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by A1, FINSEQ_5: 55;

      hence thesis by FINSEQ_6: 90, SPRECT_2: 43;

    end;

    theorem :: JORDAN1G:40

    

     Th40: for C be compact non vertical non horizontal Subset of ( TOP-REAL 2) holds ( Upper_Seq (C,n)) is_a_h.c._for ( Cage (C,n))

    proof

      let C be compact non vertical non horizontal Subset of ( TOP-REAL 2);

      

       A1: ((( Upper_Seq (C,n)) /. 1) `1 ) = (( W-min ( L~ ( Cage (C,n)))) `1 ) by JORDAN1F: 5

      .= ( W-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

      

       A2: ((( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) `1 ) = (( E-max ( L~ ( Cage (C,n)))) `1 ) by JORDAN1F: 7

      .= ( E-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

      ( Upper_Seq (C,n)) is_in_the_area_of ( Cage (C,n)) by JORDAN1E: 17;

      hence thesis by A1, A2, SPRECT_2:def 2;

    end;

    theorem :: JORDAN1G:41

    

     Th41: for C be compact non vertical non horizontal Subset of ( TOP-REAL 2) holds ( Rev ( Lower_Seq (C,n))) is_a_h.c._for ( Cage (C,n))

    proof

      let C be compact non vertical non horizontal Subset of ( TOP-REAL 2);

      

       A1: ((( Rev ( Lower_Seq (C,n))) /. 1) `1 ) = ((( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) `1 ) by FINSEQ_5: 65

      .= (( W-min ( L~ ( Cage (C,n)))) `1 ) by JORDAN1F: 8

      .= ( W-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

      

       A2: ((( Rev ( Lower_Seq (C,n))) /. ( len ( Rev ( Lower_Seq (C,n))))) `1 ) = ((( Rev ( Lower_Seq (C,n))) /. ( len ( Lower_Seq (C,n)))) `1 ) by FINSEQ_5:def 3

      .= ((( Lower_Seq (C,n)) /. 1) `1 ) by FINSEQ_5: 65

      .= (( E-max ( L~ ( Cage (C,n)))) `1 ) by JORDAN1F: 6

      .= ( E-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

      ( Rev ( Lower_Seq (C,n))) is_in_the_area_of ( Cage (C,n)) by JORDAN1E: 18, SPRECT_3: 51;

      hence thesis by A1, A2, SPRECT_2:def 2;

    end;

    theorem :: JORDAN1G:42

    

     Th42: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i be Nat st 1 < i & i <= ( len ( Gauge (C,n))) holds not (( Gauge (C,n)) * (i,1)) in ( rng ( Upper_Seq (C,n)))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i be Nat;

      assume that

       A1: 1 < i & i <= ( len ( Gauge (C,n))) and

       A2: (( Gauge (C,n)) * (i,1)) in ( rng ( Upper_Seq (C,n)));

      consider i2 be Nat such that

       A3: i2 in ( dom ( Upper_Seq (C,n))) and

       A4: (( Upper_Seq (C,n)) . i2) = (( Gauge (C,n)) * (i,1)) by A2, FINSEQ_2: 10;

      reconsider i2 as Nat;

      

       A5: 1 <= i2 & i2 <= ( len ( Upper_Seq (C,n))) by A3, FINSEQ_3: 25;

      set f = ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))));

      set i1 = (( N-min ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n)));

      

       A6: ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) & ( rng f) = ( rng ( Cage (C,n))) by FINSEQ_6: 90, SPRECT_2: 43, SPRECT_2: 46;

      ( W-min ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 43;

      then

       A7: (f /. 1) = ( W-min ( L~ ( Cage (C,n)))) by FINSEQ_6: 92;

      ( L~ ( Cage (C,n))) = ( L~ f) by REVROT_1: 33;

      then

       A8: (( N-min ( L~ ( Cage (C,n)))) .. f) < (( N-max ( L~ ( Cage (C,n)))) .. f) & (( N-max ( L~ ( Cage (C,n)))) .. f) <= (( E-max ( L~ ( Cage (C,n)))) .. f) by A7, SPRECT_5: 24, SPRECT_5: 25;

      (( E-max ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n))) = ( len ( Upper_Seq (C,n))) by Th24;

      then (( N-max ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n))) <= ( len ( Upper_Seq (C,n))) by Th23;

      then

       A9: i1 < ( len ( Upper_Seq (C,n))) by Th22, XXREAL_0: 2;

      3 <= ( len ( Lower_Seq (C,n))) by JORDAN1E: 15;

      then

       A10: 2 <= ( len ( Lower_Seq (C,n))) by XXREAL_0: 2;

      

       A11: ( len ( Gauge (C,n))) = ( width ( Gauge (C,n))) by JORDAN8:def 1;

      4 <= ( len ( Gauge (C,n))) by JORDAN8: 10;

      then

       A12: 1 <= ( len ( Gauge (C,n))) by XXREAL_0: 2;

      (( W-min ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n))) = 1 & (( W-max ( L~ ( Cage (C,n)))) .. ( Upper_Seq (C,n))) <= i1 by Th19, Th21;

      then

       A13: i1 > 1 by Th20, XXREAL_0: 2;

      then

       A14: i1 in ( dom ( Upper_Seq (C,n))) by A9, FINSEQ_3: 25;

      ( Upper_Seq (C,n)) = (f -: ( E-max ( L~ ( Cage (C,n))))) & ( N-min ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by JORDAN1E:def 1, SPRECT_2: 39;

      then

       A15: ( N-min ( L~ ( Cage (C,n)))) in ( rng ( Upper_Seq (C,n))) by A6, A8, FINSEQ_5: 46, XXREAL_0: 2;

      then

       A16: (( Upper_Seq (C,n)) /. i1) = ( N-min ( L~ ( Cage (C,n)))) by FINSEQ_5: 38;

      

       A17: i1 in NAT & i2 in NAT by ORDINAL1:def 12;

      

       A18: i1 <> i2

      proof

        assume i1 = i2;

        then (( Gauge (C,n)) * (i,1)) = ( N-min ( L~ ( Cage (C,n)))) by A4, A14, A16, PARTFUN1:def 6;

        then ((( Gauge (C,n)) * (i,1)) `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

        then ( S-bound ( L~ ( Cage (C,n)))) = ( N-bound ( L~ ( Cage (C,n)))) by A1, JORDAN1A: 72;

        hence contradiction by SPRECT_1: 16;

      end;

      then ( mid (( Upper_Seq (C,n)),i1,i2)) is being_S-Seq by A13, A9, A5, JORDAN3: 6, A17;

      then

      reconsider h1 = ( mid (( Upper_Seq (C,n)),i1,i2)) as one-to-one special FinSequence of ( TOP-REAL 2);

      set h = ( Rev h1);

      

       A19: ( len h1) = ( len h) by FINSEQ_5:def 3;

      then

       A20: h1 is non empty by A3, A14, SPRECT_2: 5;

      

      then

       A21: ((h /. ( len h)) `2 ) = ((h1 /. 1) `2 ) by A19, FINSEQ_5: 65

      .= ((( Upper_Seq (C,n)) /. i1) `2 ) by A3, A14, SPRECT_2: 8

      .= (( N-min ( L~ ( Cage (C,n)))) `2 ) by A15, FINSEQ_5: 38

      .= ( N-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

      h1 is_in_the_area_of ( Cage (C,n)) by A3, A14, JORDAN1E: 17, SPRECT_2: 22;

      then

       A22: h is_in_the_area_of ( Cage (C,n)) by SPRECT_3: 51;

      ((h /. 1) `2 ) = ((h1 /. ( len h1)) `2 ) by A20, FINSEQ_5: 65

      .= ((( Upper_Seq (C,n)) /. i2) `2 ) by A3, A14, SPRECT_2: 9

      .= ((( Gauge (C,n)) * (i,1)) `2 ) by A3, A4, PARTFUN1:def 6

      .= ( S-bound ( L~ ( Cage (C,n)))) by A1, JORDAN1A: 72;

      then

       A23: ( Rev ( Lower_Seq (C,n))) is special & h is_a_v.c._for ( Cage (C,n)) by A22, A21, SPRECT_2:def 3;

      ( len h) >= 1 by A3, A14, A19, SPRECT_2: 5;

      then ( len h) > 1 by A3, A14, A18, A19, SPRECT_2: 6, XXREAL_0: 1;

      then

       A24: (1 + 1) <= ( len h) by NAT_1: 13;

      ( len ( Lower_Seq (C,n))) = ( len ( Rev ( Lower_Seq (C,n)))) & h is special by FINSEQ_5:def 3, SPPOL_2: 40;

      then ( L~ ( Rev ( Lower_Seq (C,n)))) = ( L~ ( Lower_Seq (C,n))) & ( L~ ( Rev ( Lower_Seq (C,n)))) meets ( L~ h) by A10, A24, A23, Th41, SPPOL_2: 22, SPRECT_2: 29;

      then

      consider x be object such that

       A25: x in ( L~ ( Lower_Seq (C,n))) and

       A26: x in ( L~ h) by XBOOLE_0: 3;

      

       A27: ( L~ h) = ( L~ h1) by SPPOL_2: 22;

      ( L~ ( mid (( Upper_Seq (C,n)),i1,i2))) c= ( L~ ( Upper_Seq (C,n))) by A13, A9, A5, JORDAN4: 35;

      then x in (( L~ ( Upper_Seq (C,n))) /\ ( L~ ( Lower_Seq (C,n)))) by A25, A26, A27, XBOOLE_0:def 4;

      then

       A28: x in {( W-min ( L~ ( Cage (C,n)))), ( E-max ( L~ ( Cage (C,n))))} by JORDAN1E: 16;

      per cases by A28, TARSKI:def 2;

        suppose x = ( W-min ( L~ ( Cage (C,n))));

        then x = (( Upper_Seq (C,n)) /. 1) by JORDAN1F: 5;

        then i2 = 1 by A13, A9, A5, A26, A27, Th37;

        then (( Upper_Seq (C,n)) /. 1) = (( Gauge (C,n)) * (i,1)) by A3, A4, PARTFUN1:def 6;

        then ( W-min ( L~ ( Cage (C,n)))) = (( Gauge (C,n)) * (i,1)) by JORDAN1F: 5;

        

        then ((( Gauge (C,n)) * (i,1)) `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by EUCLID: 52

        .= ((( Gauge (C,n)) * (1,1)) `1 ) by A12, JORDAN1A: 73;

        hence contradiction by A1, A12, A11, GOBOARD5: 3;

      end;

        suppose x = ( E-max ( L~ ( Cage (C,n))));

        then x = (( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) by JORDAN1F: 7;

        then i2 = ( len ( Upper_Seq (C,n))) by A13, A9, A5, A26, A27, Th38;

        then (( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) = (( Gauge (C,n)) * (i,1)) by A3, A4, PARTFUN1:def 6;

        then

         A29: ( E-max ( L~ ( Cage (C,n)))) = (( Gauge (C,n)) * (i,1)) by JORDAN1F: 7;

        (( SE-corner ( L~ ( Cage (C,n)))) `2 ) <= (( E-min ( L~ ( Cage (C,n)))) `2 ) by PSCOMP_1: 46;

        then (( SE-corner ( L~ ( Cage (C,n)))) `2 ) < (( E-max ( L~ ( Cage (C,n)))) `2 ) by SPRECT_2: 53, XXREAL_0: 2;

        then ( S-bound ( L~ ( Cage (C,n)))) < ((( Gauge (C,n)) * (i,1)) `2 ) by A29, EUCLID: 52;

        hence contradiction by A1, JORDAN1A: 72;

      end;

    end;

    theorem :: JORDAN1G:43

    

     Th43: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i be Nat st 1 <= i & i < ( len ( Gauge (C,n))) holds not (( Gauge (C,n)) * (i,( width ( Gauge (C,n))))) in ( rng ( Lower_Seq (C,n)))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i be Nat;

      set wi = ( width ( Gauge (C,n)));

      assume that

       A1: 1 <= i & i < ( len ( Gauge (C,n))) and

       A2: (( Gauge (C,n)) * (i,wi)) in ( rng ( Lower_Seq (C,n)));

      consider i2 be Nat such that

       A3: i2 in ( dom ( Lower_Seq (C,n))) and

       A4: (( Lower_Seq (C,n)) . i2) = (( Gauge (C,n)) * (i,wi)) by A2, FINSEQ_2: 10;

      reconsider i2 as Nat;

      

       A5: 1 <= i2 & i2 <= ( len ( Lower_Seq (C,n))) by A3, FINSEQ_3: 25;

      3 <= ( len ( Upper_Seq (C,n))) by JORDAN1E: 15;

      then

       A6: 2 <= ( len ( Upper_Seq (C,n))) by XXREAL_0: 2;

      set f = ( Rotate (( Cage (C,n)),( E-max ( L~ ( Cage (C,n))))));

      set i1 = (( S-max ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n)));

      

       A7: ( len ( Gauge (C,n))) = ( width ( Gauge (C,n))) by JORDAN8:def 1;

      ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 46;

      then

       A8: (f /. 1) = ( E-max ( L~ ( Cage (C,n)))) by FINSEQ_6: 92;

      ( L~ ( Cage (C,n))) = ( L~ f) by REVROT_1: 33;

      then

       A9: (( S-max ( L~ ( Cage (C,n)))) .. f) < (( S-min ( L~ ( Cage (C,n)))) .. f) & (( S-min ( L~ ( Cage (C,n)))) .. f) <= (( W-min ( L~ ( Cage (C,n)))) .. f) by A8, SPRECT_5: 40, SPRECT_5: 41;

      

       A10: ( W-min ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) & ( rng f) = ( rng ( Cage (C,n))) by FINSEQ_6: 90, SPRECT_2: 43, SPRECT_2: 46;

      (( W-min ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n))) = ( len ( Lower_Seq (C,n))) by Th30;

      then (( S-min ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n))) <= ( len ( Lower_Seq (C,n))) by Th29;

      then

       A11: i1 < ( len ( Lower_Seq (C,n))) by Th28, XXREAL_0: 2;

      (( E-max ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n))) = 1 & (( E-min ( L~ ( Cage (C,n)))) .. ( Lower_Seq (C,n))) <= i1 by Th25, Th27;

      then

       A12: i1 > 1 by Th26, XXREAL_0: 2;

      then

       A13: i1 in ( dom ( Lower_Seq (C,n))) by A11, FINSEQ_3: 25;

      ( Lower_Seq (C,n)) = (f -: ( W-min ( L~ ( Cage (C,n))))) & ( S-max ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by Th18, SPRECT_2: 42;

      then

       A14: ( S-max ( L~ ( Cage (C,n)))) in ( rng ( Lower_Seq (C,n))) by A10, A9, FINSEQ_5: 46, XXREAL_0: 2;

      then

       A15: (( Lower_Seq (C,n)) /. i1) = ( S-max ( L~ ( Cage (C,n)))) by FINSEQ_5: 38;

      

       A16: i1 in NAT & i2 in NAT by ORDINAL1:def 12;

      

       A17: i1 <> i2

      proof

        assume i1 = i2;

        then (( Gauge (C,n)) * (i,wi)) = ( S-max ( L~ ( Cage (C,n)))) by A4, A13, A15, PARTFUN1:def 6;

        then ((( Gauge (C,n)) * (i,wi)) `2 ) = ( S-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

        then ( N-bound ( L~ ( Cage (C,n)))) = ( S-bound ( L~ ( Cage (C,n)))) by A1, A7, JORDAN1A: 70;

        hence contradiction by SPRECT_1: 16;

      end;

      then ( mid (( Lower_Seq (C,n)),i1,i2)) is being_S-Seq by A12, A11, A5, JORDAN3: 6, A16;

      then

      reconsider h = ( mid (( Lower_Seq (C,n)),i1,i2)) as one-to-one special FinSequence of ( TOP-REAL 2);

      

       A18: ((h /. 1) `2 ) = ((( Lower_Seq (C,n)) /. i1) `2 ) by A3, A13, SPRECT_2: 8

      .= (( S-max ( L~ ( Cage (C,n)))) `2 ) by A14, FINSEQ_5: 38

      .= ( S-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

      ( len h) >= 1 by A3, A13, SPRECT_2: 5;

      then ( len h) > 1 by A3, A13, A17, SPRECT_2: 6, XXREAL_0: 1;

      then

       A19: (1 + 1) <= ( len h) by NAT_1: 13;

      

       A20: h is_in_the_area_of ( Cage (C,n)) by A3, A13, JORDAN1E: 18, SPRECT_2: 22;

      ((h /. ( len h)) `2 ) = ((( Lower_Seq (C,n)) /. i2) `2 ) by A3, A13, SPRECT_2: 9

      .= ((( Gauge (C,n)) * (i,wi)) `2 ) by A3, A4, PARTFUN1:def 6

      .= ( N-bound ( L~ ( Cage (C,n)))) by A1, A7, JORDAN1A: 70;

      then h is_a_v.c._for ( Cage (C,n)) by A20, A18, SPRECT_2:def 3;

      then ( L~ ( Upper_Seq (C,n))) meets ( L~ h) by A6, A19, Th40, SPRECT_2: 29;

      then

      consider x be object such that

       A21: x in ( L~ ( Upper_Seq (C,n))) and

       A22: x in ( L~ h) by XBOOLE_0: 3;

      ( L~ ( mid (( Lower_Seq (C,n)),i1,i2))) c= ( L~ ( Lower_Seq (C,n))) by A12, A11, A5, JORDAN4: 35;

      then x in (( L~ ( Lower_Seq (C,n))) /\ ( L~ ( Upper_Seq (C,n)))) by A21, A22, XBOOLE_0:def 4;

      then

       A23: x in {( W-min ( L~ ( Cage (C,n)))), ( E-max ( L~ ( Cage (C,n))))} by JORDAN1E: 16;

      4 <= ( len ( Gauge (C,n))) by JORDAN8: 10;

      then

       A24: 1 <= ( len ( Gauge (C,n))) by XXREAL_0: 2;

      per cases by A23, TARSKI:def 2;

        suppose x = ( E-max ( L~ ( Cage (C,n))));

        then x = (( Lower_Seq (C,n)) /. 1) by JORDAN1F: 6;

        then i2 = 1 by A12, A11, A5, A22, Th37;

        then (( Lower_Seq (C,n)) /. 1) = (( Gauge (C,n)) * (i,wi)) by A3, A4, PARTFUN1:def 6;

        then ( E-max ( L~ ( Cage (C,n)))) = (( Gauge (C,n)) * (i,wi)) by JORDAN1F: 6;

        

        then ((( Gauge (C,n)) * (i,wi)) `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by EUCLID: 52

        .= ((( Gauge (C,n)) * (( len ( Gauge (C,n))),wi)) `1 ) by A7, A24, JORDAN1A: 71;

        hence contradiction by A1, A7, A24, GOBOARD5: 3;

      end;

        suppose x = ( W-min ( L~ ( Cage (C,n))));

        then x = (( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) by JORDAN1F: 8;

        then i2 = ( len ( Lower_Seq (C,n))) by A12, A11, A5, A22, Th38;

        then (( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) = (( Gauge (C,n)) * (i,wi)) by A3, A4, PARTFUN1:def 6;

        then

         A25: ( W-min ( L~ ( Cage (C,n)))) = (( Gauge (C,n)) * (i,wi)) by JORDAN1F: 8;

        (( NW-corner ( L~ ( Cage (C,n)))) `2 ) >= (( W-max ( L~ ( Cage (C,n)))) `2 ) by PSCOMP_1: 30;

        then (( NW-corner ( L~ ( Cage (C,n)))) `2 ) > (( W-min ( L~ ( Cage (C,n)))) `2 ) by SPRECT_2: 57, XXREAL_0: 2;

        then ( N-bound ( L~ ( Cage (C,n)))) > ((( Gauge (C,n)) * (i,wi)) `2 ) by A25, EUCLID: 52;

        hence contradiction by A1, A7, JORDAN1A: 70;

      end;

    end;

    theorem :: JORDAN1G:44

    

     Th44: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i be Nat st 1 < i & i <= ( len ( Gauge (C,n))) holds not (( Gauge (C,n)) * (i,1)) in ( L~ ( Upper_Seq (C,n)))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i be Nat such that

       A1: 1 < i & i <= ( len ( Gauge (C,n))) and

       A2: (( Gauge (C,n)) * (i,1)) in ( L~ ( Upper_Seq (C,n)));

      set Gi1 = (( Gauge (C,n)) * (i,1));

      consider ii be Nat such that

       A3: 1 <= ii and

       A4: (ii + 1) <= ( len ( Upper_Seq (C,n))) and

       A5: Gi1 in ( LSeg (( Upper_Seq (C,n)),ii)) by A2, SPPOL_2: 13;

      

       A6: ( LSeg (( Upper_Seq (C,n)),ii)) = ( LSeg ((( Upper_Seq (C,n)) /. ii),(( Upper_Seq (C,n)) /. (ii + 1)))) by A3, A4, TOPREAL1:def 3;

      (ii + 1) >= 1 by NAT_1: 11;

      then

       A7: (ii + 1) in ( dom ( Upper_Seq (C,n))) by A4, FINSEQ_3: 25;

      ( len ( Gauge (C,n))) >= 4 by JORDAN8: 10;

      then ( len ( Gauge (C,n))) = ( width ( Gauge (C,n))) & ( len ( Gauge (C,n))) > 1 by JORDAN8:def 1, XXREAL_0: 2;

      then

       A8: [i, 1] in ( Indices ( Gauge (C,n))) by A1, MATRIX_0: 30;

      ii < ( len ( Upper_Seq (C,n))) by A4, NAT_1: 13;

      then

       A9: ii in ( dom ( Upper_Seq (C,n))) by A3, FINSEQ_3: 25;

      

       A10: not Gi1 in ( rng ( Upper_Seq (C,n))) by A1, Th42;

      ( Upper_Seq (C,n)) is_sequence_on ( Gauge (C,n)) by Th4;

      then

      consider i1,j1,i2,j2 be Nat such that

       A11: [i1, j1] in ( Indices ( Gauge (C,n))) and

       A12: (( Upper_Seq (C,n)) /. ii) = (( Gauge (C,n)) * (i1,j1)) and

       A13: [i2, j2] in ( Indices ( Gauge (C,n))) and

       A14: (( Upper_Seq (C,n)) /. (ii + 1)) = (( Gauge (C,n)) * (i2,j2)) and

       A15: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A3, A4, JORDAN8: 3;

      

       A16: 1 <= i1 by A11, MATRIX_0: 32;

      

       A17: j2 <= ( width ( Gauge (C,n))) by A13, MATRIX_0: 32;

      

       A18: 1 <= j1 by A11, MATRIX_0: 32;

      

       A19: i1 <= ( len ( Gauge (C,n))) by A11, MATRIX_0: 32;

      

       A20: 1 <= j2 by A13, MATRIX_0: 32;

      

       A21: i2 <= ( len ( Gauge (C,n))) by A13, MATRIX_0: 32;

      

       A22: 1 <= i2 by A13, MATRIX_0: 32;

      

       A23: j1 <= ( width ( Gauge (C,n))) by A11, MATRIX_0: 32;

      per cases by A15;

        suppose

         A24: i1 = i2 & (j1 + 1) = j2;

        then j1 <= j2 by NAT_1: 11;

        then ((( Gauge (C,n)) * (i1,j1)) `2 ) <= ((( Gauge (C,n)) * (i2,j2)) `2 ) by A16, A19, A18, A17, A24, SPRECT_3: 12;

        then

         A25: ((( Gauge (C,n)) * (i1,j1)) `2 ) <= (Gi1 `2 ) by A5, A6, A12, A14, TOPREAL1: 4;

        ((( Gauge (C,n)) * (i1,j1)) `1 ) = ((( Gauge (C,n)) * (i2,1)) `1 ) by A16, A19, A18, A23, A24, GOBOARD5: 2

        .= ((( Gauge (C,n)) * (i2,j2)) `1 ) by A22, A21, A20, A17, GOBOARD5: 2;

        then ( LSeg ((( Upper_Seq (C,n)) /. ii),(( Upper_Seq (C,n)) /. (ii + 1)))) is vertical by A12, A14, SPPOL_1: 16;

        then (Gi1 `1 ) = ((( Gauge (C,n)) * (i1,j1)) `1 ) by A5, A6, A12, SPPOL_1: 41;

        then

         A26: i1 = i by A11, A8, Th7;

        then (Gi1 `2 ) <= ((( Gauge (C,n)) * (i1,j1)) `2 ) by A16, A19, A18, A23, SPRECT_3: 12;

        then j1 = 1 by A11, A8, A25, Th6, XXREAL_0: 1;

        hence contradiction by A12, A9, A10, A26, PARTFUN2: 2;

      end;

        suppose

         A27: (i1 + 1) = i2 & j1 = j2;

        

        then ((( Gauge (C,n)) * (i1,j1)) `2 ) = ((( Gauge (C,n)) * (1,j2)) `2 ) by A16, A19, A18, A23, GOBOARD5: 1

        .= ((( Gauge (C,n)) * (i2,j2)) `2 ) by A22, A21, A20, A17, GOBOARD5: 1;

        then ( LSeg ((( Upper_Seq (C,n)) /. ii),(( Upper_Seq (C,n)) /. (ii + 1)))) is horizontal by A12, A14, SPPOL_1: 15;

        then (Gi1 `2 ) = ((( Gauge (C,n)) * (i1,j1)) `2 ) by A5, A6, A12, SPPOL_1: 40;

        then

         A28: j1 = 1 by A11, A8, Th6;

        i2 > 1 by A16, A27, NAT_1: 13;

        then not (( Upper_Seq (C,n)) /. (ii + 1)) in ( rng ( Upper_Seq (C,n))) by A14, A21, A27, A28, Th42;

        hence contradiction by A7, PARTFUN2: 2;

      end;

        suppose

         A29: i1 = (i2 + 1) & j1 = j2;

        

        then ((( Gauge (C,n)) * (i1,j1)) `2 ) = ((( Gauge (C,n)) * (1,j2)) `2 ) by A16, A19, A18, A23, GOBOARD5: 1

        .= ((( Gauge (C,n)) * (i2,j2)) `2 ) by A22, A21, A20, A17, GOBOARD5: 1;

        then ( LSeg ((( Upper_Seq (C,n)) /. ii),(( Upper_Seq (C,n)) /. (ii + 1)))) is horizontal by A12, A14, SPPOL_1: 15;

        then (Gi1 `2 ) = ((( Gauge (C,n)) * (i1,j1)) `2 ) by A5, A6, A12, SPPOL_1: 40;

        then

         A30: j1 = 1 by A11, A8, Th6;

        i1 > 1 by A22, A29, NAT_1: 13;

        then not (( Upper_Seq (C,n)) /. ii) in ( rng ( Upper_Seq (C,n))) by A12, A19, A30, Th42;

        hence contradiction by A9, PARTFUN2: 2;

      end;

        suppose

         A31: i1 = i2 & j1 = (j2 + 1);

        then j2 <= j1 by NAT_1: 11;

        then ((( Gauge (C,n)) * (i2,j2)) `2 ) <= ((( Gauge (C,n)) * (i1,j1)) `2 ) by A16, A19, A23, A20, A31, SPRECT_3: 12;

        then

         A32: ((( Gauge (C,n)) * (i2,j2)) `2 ) <= (Gi1 `2 ) by A5, A6, A12, A14, TOPREAL1: 4;

        ((( Gauge (C,n)) * (i1,j1)) `1 ) = ((( Gauge (C,n)) * (i2,1)) `1 ) by A16, A19, A18, A23, A31, GOBOARD5: 2

        .= ((( Gauge (C,n)) * (i2,j2)) `1 ) by A22, A21, A20, A17, GOBOARD5: 2;

        then ( LSeg ((( Upper_Seq (C,n)) /. ii),(( Upper_Seq (C,n)) /. (ii + 1)))) is vertical by A12, A14, SPPOL_1: 16;

        then (Gi1 `1 ) = ((( Gauge (C,n)) * (i1,j1)) `1 ) by A5, A6, A12, SPPOL_1: 41;

        then

         A33: i1 = i by A11, A8, Th7;

        then (Gi1 `2 ) <= ((( Gauge (C,n)) * (i2,j2)) `2 ) by A22, A21, A20, A17, A31, SPRECT_3: 12;

        then j2 = 1 by A13, A8, A32, Th6, XXREAL_0: 1;

        hence contradiction by A14, A7, A10, A31, A33, PARTFUN2: 2;

      end;

    end;

    theorem :: JORDAN1G:45

    for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i be Nat st 1 <= i & i < ( len ( Gauge (C,n))) holds not (( Gauge (C,n)) * (i,( width ( Gauge (C,n))))) in ( L~ ( Lower_Seq (C,n)))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      set wi = ( width ( Gauge (C,n)));

      let i be Nat such that

       A1: 1 <= i & i < ( len ( Gauge (C,n))) and

       A2: (( Gauge (C,n)) * (i,wi)) in ( L~ ( Lower_Seq (C,n)));

      set Gi1 = (( Gauge (C,n)) * (i,wi));

      consider ii be Nat such that

       A3: 1 <= ii and

       A4: (ii + 1) <= ( len ( Lower_Seq (C,n))) and

       A5: Gi1 in ( LSeg (( Lower_Seq (C,n)),ii)) by A2, SPPOL_2: 13;

      

       A6: ( LSeg (( Lower_Seq (C,n)),ii)) = ( LSeg ((( Lower_Seq (C,n)) /. ii),(( Lower_Seq (C,n)) /. (ii + 1)))) by A3, A4, TOPREAL1:def 3;

      (ii + 1) >= 1 by NAT_1: 11;

      then

       A7: (ii + 1) in ( dom ( Lower_Seq (C,n))) by A4, FINSEQ_3: 25;

      ( len ( Gauge (C,n))) >= 4 by JORDAN8: 10;

      then ( len ( Gauge (C,n))) = ( width ( Gauge (C,n))) & ( len ( Gauge (C,n))) > 1 by JORDAN8:def 1, XXREAL_0: 2;

      then

       A8: [i, wi] in ( Indices ( Gauge (C,n))) by A1, MATRIX_0: 30;

      ii < ( len ( Lower_Seq (C,n))) by A4, NAT_1: 13;

      then

       A9: ii in ( dom ( Lower_Seq (C,n))) by A3, FINSEQ_3: 25;

      

       A10: not Gi1 in ( rng ( Lower_Seq (C,n))) by A1, Th43;

      ( Lower_Seq (C,n)) is_sequence_on ( Gauge (C,n)) by Th5;

      then

      consider i1,j1,i2,j2 be Nat such that

       A11: [i1, j1] in ( Indices ( Gauge (C,n))) and

       A12: (( Lower_Seq (C,n)) /. ii) = (( Gauge (C,n)) * (i1,j1)) and

       A13: [i2, j2] in ( Indices ( Gauge (C,n))) and

       A14: (( Lower_Seq (C,n)) /. (ii + 1)) = (( Gauge (C,n)) * (i2,j2)) and

       A15: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A3, A4, JORDAN8: 3;

      

       A16: 1 <= i1 by A11, MATRIX_0: 32;

      

       A17: j2 <= ( width ( Gauge (C,n))) by A13, MATRIX_0: 32;

      

       A18: 1 <= j1 by A11, MATRIX_0: 32;

      

       A19: i1 <= ( len ( Gauge (C,n))) by A11, MATRIX_0: 32;

      

       A20: 1 <= j2 by A13, MATRIX_0: 32;

      

       A21: i2 <= ( len ( Gauge (C,n))) by A13, MATRIX_0: 32;

      

       A22: 1 <= i2 by A13, MATRIX_0: 32;

      

       A23: j1 <= ( width ( Gauge (C,n))) by A11, MATRIX_0: 32;

      per cases by A15;

        suppose

         A24: i1 = i2 & (j2 + 1) = j1;

        then j1 >= j2 by NAT_1: 11;

        then ((( Gauge (C,n)) * (i1,j1)) `2 ) >= ((( Gauge (C,n)) * (i2,j2)) `2 ) by A16, A19, A23, A20, A24, SPRECT_3: 12;

        then

         A25: ((( Gauge (C,n)) * (i1,j1)) `2 ) >= (Gi1 `2 ) by A5, A6, A12, A14, TOPREAL1: 4;

        ((( Gauge (C,n)) * (i1,j1)) `1 ) = ((( Gauge (C,n)) * (i2,1)) `1 ) by A16, A19, A18, A23, A24, GOBOARD5: 2

        .= ((( Gauge (C,n)) * (i2,j2)) `1 ) by A22, A21, A20, A17, GOBOARD5: 2;

        then ( LSeg ((( Lower_Seq (C,n)) /. ii),(( Lower_Seq (C,n)) /. (ii + 1)))) is vertical by A12, A14, SPPOL_1: 16;

        then (Gi1 `1 ) = ((( Gauge (C,n)) * (i1,j1)) `1 ) by A5, A6, A12, SPPOL_1: 41;

        then

         A26: i1 = i by A11, A8, Th7;

        then (Gi1 `2 ) >= ((( Gauge (C,n)) * (i1,j1)) `2 ) by A16, A19, A18, A23, SPRECT_3: 12;

        then j1 = wi by A11, A8, A25, Th6, XXREAL_0: 1;

        hence contradiction by A12, A9, A10, A26, PARTFUN2: 2;

      end;

        suppose

         A27: (i2 + 1) = i1 & j1 = j2;

        

        then ((( Gauge (C,n)) * (i1,j1)) `2 ) = ((( Gauge (C,n)) * (1,j2)) `2 ) by A16, A19, A18, A23, GOBOARD5: 1

        .= ((( Gauge (C,n)) * (i2,j2)) `2 ) by A22, A21, A20, A17, GOBOARD5: 1;

        then ( LSeg ((( Lower_Seq (C,n)) /. ii),(( Lower_Seq (C,n)) /. (ii + 1)))) is horizontal by A12, A14, SPPOL_1: 15;

        then (Gi1 `2 ) = ((( Gauge (C,n)) * (i1,j1)) `2 ) by A5, A6, A12, SPPOL_1: 40;

        then

         A28: j1 = wi by A11, A8, Th6;

        i2 < ( len ( Gauge (C,n))) by A19, A27, NAT_1: 13;

        then not (( Lower_Seq (C,n)) /. (ii + 1)) in ( rng ( Lower_Seq (C,n))) by A14, A22, A27, A28, Th43;

        hence contradiction by A7, PARTFUN2: 2;

      end;

        suppose

         A29: i2 = (i1 + 1) & j1 = j2;

        

        then ((( Gauge (C,n)) * (i1,j1)) `2 ) = ((( Gauge (C,n)) * (1,j2)) `2 ) by A16, A19, A18, A23, GOBOARD5: 1

        .= ((( Gauge (C,n)) * (i2,j2)) `2 ) by A22, A21, A20, A17, GOBOARD5: 1;

        then ( LSeg ((( Lower_Seq (C,n)) /. ii),(( Lower_Seq (C,n)) /. (ii + 1)))) is horizontal by A12, A14, SPPOL_1: 15;

        then (Gi1 `2 ) = ((( Gauge (C,n)) * (i1,j1)) `2 ) by A5, A6, A12, SPPOL_1: 40;

        then

         A30: j1 = wi by A11, A8, Th6;

        i1 < ( len ( Gauge (C,n))) by A21, A29, NAT_1: 13;

        then not (( Lower_Seq (C,n)) /. ii) in ( rng ( Lower_Seq (C,n))) by A12, A16, A30, Th43;

        hence contradiction by A9, PARTFUN2: 2;

      end;

        suppose

         A31: i1 = i2 & j2 = (j1 + 1);

        then j2 >= j1 by NAT_1: 11;

        then ((( Gauge (C,n)) * (i2,j2)) `2 ) >= ((( Gauge (C,n)) * (i1,j1)) `2 ) by A16, A19, A18, A17, A31, SPRECT_3: 12;

        then

         A32: ((( Gauge (C,n)) * (i2,j2)) `2 ) >= (Gi1 `2 ) by A5, A6, A12, A14, TOPREAL1: 4;

        ((( Gauge (C,n)) * (i1,j1)) `1 ) = ((( Gauge (C,n)) * (i2,1)) `1 ) by A16, A19, A18, A23, A31, GOBOARD5: 2

        .= ((( Gauge (C,n)) * (i2,j2)) `1 ) by A22, A21, A20, A17, GOBOARD5: 2;

        then ( LSeg ((( Lower_Seq (C,n)) /. ii),(( Lower_Seq (C,n)) /. (ii + 1)))) is vertical by A12, A14, SPPOL_1: 16;

        then (Gi1 `1 ) = ((( Gauge (C,n)) * (i1,j1)) `1 ) by A5, A6, A12, SPPOL_1: 41;

        then

         A33: i1 = i by A11, A8, Th7;

        then (Gi1 `2 ) >= ((( Gauge (C,n)) * (i2,j2)) `2 ) by A22, A21, A20, A17, A31, SPRECT_3: 12;

        then j2 = wi by A13, A8, A32, Th6, XXREAL_0: 1;

        hence contradiction by A14, A7, A10, A31, A33, PARTFUN2: 2;

      end;

    end;

    theorem :: JORDAN1G:46

    

     Th46: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for i,j be Nat st 1 <= i & i <= ( len ( Gauge (C,n))) & 1 <= j & j <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (i,j)) in ( L~ ( Cage (C,n))) holds ( LSeg ((( Gauge (C,n)) * (i,1)),(( Gauge (C,n)) * (i,j)))) meets ( L~ ( Lower_Seq (C,n)))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let i,j be Nat;

      set Gij = (( Gauge (C,n)) * (i,j));

      assume that

       A1: 1 <= i and

       A2: i <= ( len ( Gauge (C,n))) and

       A3: 1 <= j & j <= ( width ( Gauge (C,n))) and

       A4: Gij in ( L~ ( Cage (C,n)));

      

       A5: ( Lower_Seq (C,n)) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) :- ( E-max ( L~ ( Cage (C,n))))) by JORDAN1E:def 2;

      set Wmi = ( W-min ( L~ ( Cage (C,n))));

      set h = ( mid (( Lower_Seq (C,n)),2,( len ( Lower_Seq (C,n)))));

      set v1 = ( L_Cut (( Upper_Seq (C,n)),Gij));

      set NE = ( NE-corner ( L~ ( Cage (C,n))));

      set Gv1 = ( <*(( Gauge (C,n)) * (i,1))*> ^ v1);

      set v = (Gv1 ^ <*NE*>);

      

       A6: ( L~ ( Cage (C,n))) = (( L~ ( Upper_Seq (C,n))) \/ ( L~ ( Lower_Seq (C,n)))) by JORDAN1E: 13;

      

       A7: ( Upper_Seq (C,n)) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) -: ( E-max ( L~ ( Cage (C,n))))) by JORDAN1E:def 1;

      

       A8: ( len ( Upper_Seq (C,n))) >= 3 by JORDAN1E: 15;

      then

       A9: ( len ( Upper_Seq (C,n))) >= 1 by XXREAL_0: 2;

      

       A10: ( len ( Lower_Seq (C,n))) >= 3 by JORDAN1E: 15;

      then

       A11: ( len ( Lower_Seq (C,n))) >= 2 & ( len ( Lower_Seq (C,n))) >= 1 by XXREAL_0: 2;

      

       A12: ( len ( Gauge (C,n))) = ( width ( Gauge (C,n))) by JORDAN8:def 1;

      

       A13: ((( Gauge (C,n)) * (i,1)) `2 ) = ( S-bound ( L~ ( Cage (C,n)))) by A1, A2, JORDAN1A: 72;

      now

        per cases by A1, A4, A6, XBOOLE_0:def 3, XXREAL_0: 1;

          suppose

           A14: Gij in ( L~ ( Upper_Seq (C,n))) & i = 1;

          set G11 = (( Gauge (C,n)) * (1,1));

          

           A15: Wmi in ( L~ ( Cage (C,n))) by SPRECT_1: 13;

          ( S-bound ( L~ ( Cage (C,n)))) = (G11 `2 ) by A2, A14, JORDAN1A: 72;

          then

           A16: (Wmi `1 ) = ( W-bound ( L~ ( Cage (C,n)))) & (G11 `2 ) <= (Wmi `2 ) by A15, EUCLID: 52, PSCOMP_1: 24;

          

           A17: ( rng ( Lower_Seq (C,n))) c= ( L~ ( Lower_Seq (C,n))) by A10, SPPOL_2: 18, XXREAL_0: 2;

          

           A18: (Gij `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by A3, A12, A14, JORDAN1A: 73;

          then Gij in ( W-most ( L~ ( Cage (C,n)))) by A4, SPRECT_2: 12;

          then

           A19: (Wmi `2 ) <= (Gij `2 ) by PSCOMP_1: 31;

          (( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) = Wmi by JORDAN1F: 8;

          then

           A20: Wmi in ( rng ( Lower_Seq (C,n))) by FINSEQ_6: 168;

          (G11 `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by A2, A14, JORDAN1A: 73;

          then Wmi in ( LSeg ((( Gauge (C,n)) * (1,1)),(( Gauge (C,n)) * (1,j)))) by A14, A16, A18, A19, GOBOARD7: 7;

          hence thesis by A14, A17, A20, XBOOLE_0: 3;

        end;

          suppose

           A21: Gij in ( L~ ( Upper_Seq (C,n))) & Gij <> (( Upper_Seq (C,n)) . ( len ( Upper_Seq (C,n)))) & ( E-max ( L~ ( Cage (C,n)))) <> NE & i > 1;

          ( len ( Cage (C,n))) > 4 by GOBOARD7: 34;

          then

           A22: ( rng ( Cage (C,n))) c= ( L~ ( Cage (C,n))) by SPPOL_2: 18, XXREAL_0: 2;

          

           A23: not NE in ( rng ( Cage (C,n)))

          proof

            

             A24: (NE `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

            then (NE `1 ) = ( E-bound ( L~ ( Cage (C,n)))) & (NE `2 ) >= ( S-bound ( L~ ( Cage (C,n)))) by EUCLID: 52, SPRECT_1: 22;

            then NE in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = ( E-bound ( L~ ( Cage (C,n)))) & (p `2 ) <= ( N-bound ( L~ ( Cage (C,n)))) & (p `2 ) >= ( S-bound ( L~ ( Cage (C,n)))) } by A24;

            then

             A25: NE in ( LSeg (( SE-corner ( L~ ( Cage (C,n)))),( NE-corner ( L~ ( Cage (C,n)))))) by SPRECT_1: 23;

            assume NE in ( rng ( Cage (C,n)));

            then NE in (( LSeg (( SE-corner ( L~ ( Cage (C,n)))),( NE-corner ( L~ ( Cage (C,n)))))) /\ ( L~ ( Cage (C,n)))) by A22, A25, XBOOLE_0:def 4;

            then

             A26: (NE `2 ) <= (( E-max ( L~ ( Cage (C,n)))) `2 ) by PSCOMP_1: 47;

            

             A27: (( E-max ( L~ ( Cage (C,n)))) `1 ) = (NE `1 ) by PSCOMP_1: 45;

            (( E-max ( L~ ( Cage (C,n)))) `2 ) <= (NE `2 ) by PSCOMP_1: 46;

            then (( E-max ( L~ ( Cage (C,n)))) `2 ) = (NE `2 ) by A26, XXREAL_0: 1;

            hence contradiction by A21, A27, TOPREAL3: 6;

          end;

           A28:

          now

            per cases ;

              suppose Gij <> (( Upper_Seq (C,n)) . (( Index (Gij,( Upper_Seq (C,n)))) + 1));

              then v1 = ( <*Gij*> ^ ( mid (( Upper_Seq (C,n)),(( Index (Gij,( Upper_Seq (C,n)))) + 1),( len ( Upper_Seq (C,n)))))) by JORDAN3:def 3;

              then ( rng v1) = (( rng <*Gij*>) \/ ( rng ( mid (( Upper_Seq (C,n)),(( Index (Gij,( Upper_Seq (C,n)))) + 1),( len ( Upper_Seq (C,n))))))) by FINSEQ_1: 31;

              then

               A29: ( rng v1) = ( {Gij} \/ ( rng ( mid (( Upper_Seq (C,n)),(( Index (Gij,( Upper_Seq (C,n)))) + 1),( len ( Upper_Seq (C,n))))))) by FINSEQ_1: 38;

               not NE in ( L~ ( Cage (C,n)))

              proof

                assume NE in ( L~ ( Cage (C,n)));

                then

                consider i be Nat such that

                 A30: 1 <= i and

                 A31: (i + 1) <= ( len ( Cage (C,n))) and

                 A32: NE in ( LSeg ((( Cage (C,n)) /. i),(( Cage (C,n)) /. (i + 1)))) by SPPOL_2: 14;

                per cases by A30, A31, TOPREAL1:def 5;

                  suppose

                   A33: ((( Cage (C,n)) /. i) `1 ) = ((( Cage (C,n)) /. (i + 1)) `1 );

                  ((( Cage (C,n)) /. i) `2 ) <= ((( Cage (C,n)) /. (i + 1)) `2 ) or ((( Cage (C,n)) /. i) `2 ) >= ((( Cage (C,n)) /. (i + 1)) `2 );

                  then

                   A34: (NE `2 ) <= ((( Cage (C,n)) /. (i + 1)) `2 ) or (NE `2 ) <= ((( Cage (C,n)) /. i) `2 ) by A32, TOPREAL1: 4;

                  

                   A35: (NE `1 ) = ((( Cage (C,n)) /. i) `1 ) by A32, A33, GOBOARD7: 5;

                  

                   A36: 1 <= (i + 1) by NAT_1: 11;

                  then

                   A37: (i + 1) in ( dom ( Cage (C,n))) by A31, FINSEQ_3: 25;

                  

                   A38: (NE `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

                  then

                   A39: ((( Cage (C,n)) /. (i + 1)) `2 ) <= (NE `2 ) by A31, A36, JORDAN5D: 11;

                  

                   A40: i < ( len ( Cage (C,n))) by A31, NAT_1: 13;

                  then ((( Cage (C,n)) /. i) `2 ) <= (NE `2 ) by A30, A38, JORDAN5D: 11;

                  then (NE `2 ) = ((( Cage (C,n)) /. (i + 1)) `2 ) or (NE `2 ) = ((( Cage (C,n)) /. i) `2 ) by A39, A34, XXREAL_0: 1;

                  then

                   A41: NE = (( Cage (C,n)) /. (i + 1)) or NE = (( Cage (C,n)) /. i) by A33, A35, TOPREAL3: 6;

                  i in ( dom ( Cage (C,n))) by A30, A40, FINSEQ_3: 25;

                  hence contradiction by A23, A37, A41, PARTFUN2: 2;

                end;

                  suppose

                   A42: ((( Cage (C,n)) /. i) `2 ) = ((( Cage (C,n)) /. (i + 1)) `2 );

                  ((( Cage (C,n)) /. i) `1 ) <= ((( Cage (C,n)) /. (i + 1)) `1 ) or ((( Cage (C,n)) /. i) `1 ) >= ((( Cage (C,n)) /. (i + 1)) `1 );

                  then

                   A43: (NE `1 ) <= ((( Cage (C,n)) /. (i + 1)) `1 ) or (NE `1 ) <= ((( Cage (C,n)) /. i) `1 ) by A32, TOPREAL1: 3;

                  

                   A44: (NE `2 ) = ((( Cage (C,n)) /. i) `2 ) by A32, A42, GOBOARD7: 6;

                  

                   A45: 1 <= (i + 1) by NAT_1: 11;

                  then

                   A46: (i + 1) in ( dom ( Cage (C,n))) by A31, FINSEQ_3: 25;

                  

                   A47: (NE `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

                  then

                   A48: ((( Cage (C,n)) /. (i + 1)) `1 ) <= (NE `1 ) by A31, A45, JORDAN5D: 12;

                  

                   A49: i < ( len ( Cage (C,n))) by A31, NAT_1: 13;

                  then ((( Cage (C,n)) /. i) `1 ) <= (NE `1 ) by A30, A47, JORDAN5D: 12;

                  then (NE `1 ) = ((( Cage (C,n)) /. (i + 1)) `1 ) or (NE `1 ) = ((( Cage (C,n)) /. i) `1 ) by A48, A43, XXREAL_0: 1;

                  then

                   A50: NE = (( Cage (C,n)) /. (i + 1)) or NE = (( Cage (C,n)) /. i) by A42, A44, TOPREAL3: 6;

                  i in ( dom ( Cage (C,n))) by A30, A49, FINSEQ_3: 25;

                  hence contradiction by A23, A46, A50, PARTFUN2: 2;

                end;

              end;

              then

               A51: not NE in {Gij} by A4, TARSKI:def 1;

              ( rng ( mid (( Upper_Seq (C,n)),(( Index (Gij,( Upper_Seq (C,n)))) + 1),( len ( Upper_Seq (C,n)))))) c= ( rng ( Upper_Seq (C,n))) & ( rng ( Upper_Seq (C,n))) c= ( rng ( Cage (C,n))) by Th39, FINSEQ_6: 119;

              then ( rng ( mid (( Upper_Seq (C,n)),(( Index (Gij,( Upper_Seq (C,n)))) + 1),( len ( Upper_Seq (C,n)))))) c= ( rng ( Cage (C,n)));

              then not NE in ( rng ( mid (( Upper_Seq (C,n)),(( Index (Gij,( Upper_Seq (C,n)))) + 1),( len ( Upper_Seq (C,n)))))) by A23;

              hence not NE in ( rng v1) by A29, A51, XBOOLE_0:def 3;

            end;

              suppose Gij = (( Upper_Seq (C,n)) . (( Index (Gij,( Upper_Seq (C,n)))) + 1));

              then v1 = ( mid (( Upper_Seq (C,n)),(( Index (Gij,( Upper_Seq (C,n)))) + 1),( len ( Upper_Seq (C,n))))) by JORDAN3:def 3;

              then

               A52: ( rng v1) c= ( rng ( Upper_Seq (C,n))) by FINSEQ_6: 119;

              ( rng ( Upper_Seq (C,n))) c= ( rng ( Cage (C,n))) by Th39;

              then ( rng v1) c= ( rng ( Cage (C,n))) by A52;

              hence not NE in ( rng v1) by A23;

            end;

          end;

          ( S-bound ( L~ ( Cage (C,n)))) < ( N-bound ( L~ ( Cage (C,n)))) by SPRECT_1: 32;

          then NE <> (( Gauge (C,n)) * (i,1)) by A13, EUCLID: 52;

          then not NE in {(( Gauge (C,n)) * (i,1))} by TARSKI:def 1;

          then not NE in ( rng <*(( Gauge (C,n)) * (i,1))*>) by FINSEQ_1: 39;

          then not NE in (( rng <*(( Gauge (C,n)) * (i,1))*>) \/ ( rng v1)) by A28, XBOOLE_0:def 3;

          then not NE in ( rng Gv1) by FINSEQ_1: 31;

          then ( rng Gv1) misses {NE} by ZFMISC_1: 50;

          then

           A53: ( rng Gv1) misses ( rng <*NE*>) by FINSEQ_1: 38;

          

           A54: ( len v) = (( len Gv1) + 1) by FINSEQ_2: 16

          .= ((1 + ( len v1)) + 1) by FINSEQ_5: 8;

          

           A55: v1 is non empty by A21, JORDAN1E: 3;

          then

           A56: ( 0 + 1) <= ( len v1) by NAT_1: 13;

          then 1 in ( dom v1) by FINSEQ_3: 25;

          

          then

           A57: (v1 /. 1) = (v1 . 1) by PARTFUN1:def 6

          .= Gij by A21, JORDAN3: 23;

          then

           A58: ((v1 ^ <*NE*>) /. 1) = Gij by A56, BOOLMARK: 7;

          (1 + ( len v1)) >= (1 + 1) by A56, XREAL_1: 7;

          then

           A59: 2 < ( len v) by A54, NAT_1: 13;

          

           A60: v1 is being_S-Seq by A21, JORDAN3: 34;

          v = ( <*(( Gauge (C,n)) * (i,1))*> ^ (v1 ^ <*NE*>)) by FINSEQ_1: 32;

          then (v /. 1) = (( Gauge (C,n)) * (i,1)) by FINSEQ_5: 15;

          then

           A61: ((v /. 1) `2 ) = ( S-bound ( L~ ( Cage (C,n)))) by A1, A2, JORDAN1A: 72;

          ( len v) = (( len Gv1) + 1) by FINSEQ_2: 16;

          then (v /. ( len v)) = NE by FINSEQ_4: 67;

          then

           A62: ((v /. ( len v)) `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

          

           A63: (( Cage (C,n)) /. 1) = ( N-min ( L~ ( Cage (C,n)))) by JORDAN9: 32;

          then (( N-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) <= (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) by SPRECT_2: 70;

          then

           A64: (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) > 1 by A63, SPRECT_2: 69, XXREAL_0: 2;

          (( E-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) <= (( S-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) by A63, SPRECT_2: 72;

          then (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) < (( S-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) by A63, SPRECT_2: 71, XXREAL_0: 2;

          then (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) < (( S-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) by A63, SPRECT_2: 73, XXREAL_0: 2;

          then

           A65: (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) < (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) by A63, SPRECT_2: 74, XXREAL_0: 2;

          then (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) < ( len ( Cage (C,n))) by A63, SPRECT_2: 76, XXREAL_0: 2;

          then

           A66: ((( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) + 1) <= ( len ( Cage (C,n))) by NAT_1: 13;

          

           A67: ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 46;

          then (( Cage (C,n)) /. (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) = ( E-max ( L~ ( Cage (C,n)))) by FINSEQ_5: 38;

          then

           A68: ((( Cage (C,n)) /. ((( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) + 1)) `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by A64, A66, JORDAN1E: 20;

          

           A69: ( W-min ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 43;

          then

           A70: (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) = ((( len ( Cage (C,n))) + (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) - (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) by A67, A65, SPRECT_5: 9;

          now

            let m be Nat;

            assume

             A71: m in ( dom <*(( Gauge (C,n)) * (i,1))*>);

            then m in ( Seg 1) by FINSEQ_1: 38;

            then m = 1 by FINSEQ_1: 2, TARSKI:def 1;

            then ( <*(( Gauge (C,n)) * (i,1))*> . m) = (( Gauge (C,n)) * (i,1)) by FINSEQ_1: 40;

            then

             A72: ( <*(( Gauge (C,n)) * (i,1))*> /. m) = (( Gauge (C,n)) * (i,1)) by A71, PARTFUN1:def 6;

            ( width ( Gauge (C,n))) >= 4 by A12, JORDAN8: 10;

            then

             A73: 1 <= ( width ( Gauge (C,n))) by XXREAL_0: 2;

            then ((( Gauge (C,n)) * (1,1)) `1 ) <= ((( Gauge (C,n)) * (i,1)) `1 ) by A1, A2, SPRECT_3: 13;

            hence ( W-bound ( L~ ( Cage (C,n)))) <= (( <*(( Gauge (C,n)) * (i,1))*> /. m) `1 ) by A12, A72, A73, JORDAN1A: 73;

            ((( Gauge (C,n)) * (i,1)) `1 ) <= ((( Gauge (C,n)) * (( len ( Gauge (C,n))),1)) `1 ) by A1, A2, A73, SPRECT_3: 13;

            hence (( <*(( Gauge (C,n)) * (i,1))*> /. m) `1 ) <= ( E-bound ( L~ ( Cage (C,n)))) by A12, A72, A73, JORDAN1A: 71;

            thus ( S-bound ( L~ ( Cage (C,n)))) <= (( <*(( Gauge (C,n)) * (i,1))*> /. m) `2 ) by A1, A2, A72, JORDAN1A: 72;

            ( S-bound ( L~ ( Cage (C,n)))) = ((( Gauge (C,n)) * (i,1)) `2 ) by A1, A2, JORDAN1A: 72;

            hence (( <*(( Gauge (C,n)) * (i,1))*> /. m) `2 ) <= ( N-bound ( L~ ( Cage (C,n)))) by A72, SPRECT_1: 22;

          end;

          then

           A74: <*(( Gauge (C,n)) * (i,1))*> is_in_the_area_of ( Cage (C,n)) by SPRECT_2:def 1;

          

           A75: <*NE*> is_in_the_area_of ( Cage (C,n)) by SPRECT_2: 25;

          3 <= ( len ( Lower_Seq (C,n))) by JORDAN1E: 15;

          then 2 <= ( len ( Lower_Seq (C,n))) by XXREAL_0: 2;

          then

           A76: 2 in ( dom ( Lower_Seq (C,n))) by FINSEQ_3: 25;

           <*Gij*> is_in_the_area_of ( Cage (C,n)) by A21, JORDAN1E: 17, SPRECT_3: 46;

          then v1 is_in_the_area_of ( Cage (C,n)) by A21, JORDAN1E: 17, SPRECT_3: 56;

          then Gv1 is_in_the_area_of ( Cage (C,n)) by A74, SPRECT_2: 24;

          then v is_in_the_area_of ( Cage (C,n)) by A75, SPRECT_2: 24;

          then

           A77: v is_a_v.c._for ( Cage (C,n)) by A61, A62, SPRECT_2:def 3;

          

           A78: (((1 + ((( len ( Cage (C,n))) + (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) - (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n))))) + (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) - ( len ( Cage (C,n)))) = (1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))));

          

           A79: ( len ( Lower_Seq (C,n))) in ( dom ( Lower_Seq (C,n))) by FINSEQ_5: 6;

          then h is_in_the_area_of ( Cage (C,n)) by A76, JORDAN1E: 18, SPRECT_2: 22;

          then

           A80: ( Rev h) is_in_the_area_of ( Cage (C,n)) by SPRECT_3: 51;

          (1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) <= ( 0 + (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) by A65, NAT_1: 13;

          then ((1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) - (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) <= 0 by XREAL_1: 20;

          then

           A81: (( len ( Cage (C,n))) + ((1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) - (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n))))) <= (( len ( Cage (C,n))) + 0 ) by XREAL_1: 6;

          

           A82: ( len ( Lower_Seq (C,n))) >= (2 + 1) by JORDAN1E: 15;

          then

           A83: ( len ( Lower_Seq (C,n))) > 2 by NAT_1: 13;

          (( len ( Cage (C,n))) + 0 ) <= (( len ( Cage (C,n))) + (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) by XREAL_1: 6;

          then (( len ( Cage (C,n))) - (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) <= (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by A70, XREAL_1: 9;

          then ((( len ( Cage (C,n))) - (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) + 1) <= (1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))))) by XREAL_1: 6;

          then

           A84: ( len (( Cage (C,n)) :- ( W-min ( L~ ( Cage (C,n)))))) <= (1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))))) by A69, FINSEQ_5: 50;

          ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 46;

          then

           A85: ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by FINSEQ_6: 90, SPRECT_2: 43;

          

           A86: ( L~ v1) c= ( L~ ( Upper_Seq (C,n))) by A21, JORDAN3: 42;

          

           A87: ( len ( Lower_Seq (C,n))) > 1 by A82, XXREAL_0: 2;

          then

           A88: h is non empty by A83, JORDAN1B: 2;

          

           A89: ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by A67, FINSEQ_6: 90, SPRECT_2: 43;

          

          then (( Lower_Seq (C,n)) /. (1 + 1)) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) /. (1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))))) by A5, A76, FINSEQ_5: 52

          .= (( Cage (C,n)) /. (((1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))))) + (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) -' ( len ( Cage (C,n))))) by A69, A70, A84, A81, FINSEQ_6: 182

          .= (( Cage (C,n)) /. ((( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) + 1)) by A70, A78, XREAL_0:def 2;

          then ((h /. 1) `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by A76, A79, A68, SPRECT_2: 8;

          then ((( Rev h) /. ( len h)) `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by A88, FINSEQ_5: 65;

          then

           A90: ((( Rev h) /. ( len ( Rev h))) `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by FINSEQ_5:def 3;

          (( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) /. ( len ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))))) by A5, A89, FINSEQ_5: 54

          .= (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) /. 1) by FINSEQ_6:def 1

          .= ( W-min ( L~ ( Cage (C,n)))) by A69, FINSEQ_6: 92;

          then ((( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

          then ((h /. ( len h)) `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by A76, A79, SPRECT_2: 9;

          then ((( Rev h) /. 1) `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by A88, FINSEQ_5: 65;

          then

           A91: ( Rev h) is_a_h.c._for ( Cage (C,n)) by A80, A90, SPRECT_2:def 2;

          

           A92: ( len ( Upper_Seq (C,n))) in ( dom ( Upper_Seq (C,n))) by A9, FINSEQ_3: 25;

          set ci = ( mid (( Upper_Seq (C,n)),(( Index (Gij,( Upper_Seq (C,n)))) + 1),( len ( Upper_Seq (C,n)))));

          ( rng ( Upper_Seq (C,n))) c= ( L~ ( Upper_Seq (C,n))) by A8, SPPOL_2: 18, XXREAL_0: 2;

          then

           A93: not (( Gauge (C,n)) * (i,1)) in ( rng ( Upper_Seq (C,n))) by A2, A21, Th44;

           not (( Gauge (C,n)) * (i,1)) in ( L~ ( Upper_Seq (C,n))) by A2, A21, Th44;

          then not (( Gauge (C,n)) * (i,1)) in {Gij} by A21, TARSKI:def 1;

          then

           A94: not (( Gauge (C,n)) * (i,1)) in ( rng <*Gij*>) by FINSEQ_1: 38;

          now

            per cases ;

              suppose

               A95: Gij <> (( Upper_Seq (C,n)) . (( Index (Gij,( Upper_Seq (C,n)))) + 1));

              ( rng ci) c= ( rng ( Upper_Seq (C,n))) by FINSEQ_6: 119;

              then not (( Gauge (C,n)) * (i,1)) in ( rng ci) by A93;

              then not (( Gauge (C,n)) * (i,1)) in (( rng <*Gij*>) \/ ( rng ci)) by A94, XBOOLE_0:def 3;

              then not (( Gauge (C,n)) * (i,1)) in ( rng ( <*Gij*> ^ ci)) by FINSEQ_1: 31;

              hence not (( Gauge (C,n)) * (i,1)) in ( rng v1) by A95, JORDAN3:def 3;

            end;

              suppose Gij = (( Upper_Seq (C,n)) . (( Index (Gij,( Upper_Seq (C,n)))) + 1));

              then v1 = ci by JORDAN3:def 3;

              then ( rng v1) c= ( rng ( Upper_Seq (C,n))) by FINSEQ_6: 119;

              hence not (( Gauge (C,n)) * (i,1)) in ( rng v1) by A93;

            end;

          end;

          then {(( Gauge (C,n)) * (i,1))} misses ( rng v1) by ZFMISC_1: 50;

          then

           A96: ( rng <*(( Gauge (C,n)) * (i,1))*>) misses ( rng v1) by FINSEQ_1: 38;

          

           A97: <*NE*> is one-to-one by FINSEQ_3: 93;

          (( Lower_Seq (C,n)) /. 1) = ((( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) :- ( E-max ( L~ ( Cage (C,n))))) /. 1) by JORDAN1E:def 2

          .= ( E-max ( L~ ( Cage (C,n)))) by FINSEQ_5: 53;

          then

           A98: not ( E-max ( L~ ( Cage (C,n)))) in ( L~ h) by A83, JORDAN5B: 16;

           <*(( Gauge (C,n)) * (i,1))*> is one-to-one by FINSEQ_3: 93;

          then Gv1 is one-to-one by A96, A60, FINSEQ_3: 91;

          then

           A99: v is one-to-one by A53, A97, FINSEQ_3: 91;

          

           A100: ( L~ h) c= ( L~ ( Lower_Seq (C,n))) by A11, JORDAN4: 35;

          (( <*(( Gauge (C,n)) * (i,1))*> /. ( len <*(( Gauge (C,n)) * (i,1))*>)) `1 ) = (( <*(( Gauge (C,n)) * (i,1))*> /. 1) `1 ) by FINSEQ_1: 39

          .= ((( Gauge (C,n)) * (i,1)) `1 ) by FINSEQ_4: 16

          .= ((v1 /. 1) `1 ) by A1, A2, A3, A57, GOBOARD5: 2;

          then

           A101: Gv1 is special by A60, GOBOARD2: 8;

          ( len v1) in ( dom v1) by A56, FINSEQ_3: 25;

          

          then

           A102: (v1 /. ( len v1)) = (v1 . ( len v1)) by PARTFUN1:def 6

          .= (( Upper_Seq (C,n)) . ( len ( Upper_Seq (C,n)))) by A21, JORDAN1B: 4

          .= (( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) by A92, PARTFUN1:def 6

          .= ((( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) -: ( E-max ( L~ ( Cage (C,n))))) /. (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))))) by A7, A85, FINSEQ_5: 42

          .= ( E-max ( L~ ( Cage (C,n)))) by A85, FINSEQ_5: 45;

          then (Gv1 /. ( len Gv1)) = ( E-max ( L~ ( Cage (C,n)))) by A55, SPRECT_3: 1;

          

          then ((Gv1 /. ( len Gv1)) `1 ) = (NE `1 ) by PSCOMP_1: 45

          .= (( <*NE*> /. 1) `1 ) by FINSEQ_4: 16;

          then

           A103: v is special by A101, GOBOARD2: 8;

          h is S-Sequence_in_R2 by A83, A87, JORDAN3: 6;

          then

           A104: ( Rev h) is S-Sequence_in_R2;

          then 2 <= ( len ( Rev h)) by TOPREAL1:def 8;

          then ( L~ ( Rev h)) meets ( L~ v) by A59, A99, A103, A104, A91, A77, SPRECT_2: 29;

          then ( L~ h) meets ( L~ v) by SPPOL_2: 22;

          then

          consider x be object such that

           A105: x in ( L~ h) and

           A106: x in ( L~ v) by XBOOLE_0: 3;

          

           A107: ( W-min ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 43;

          ( L~ v) = ( L~ ( <*(( Gauge (C,n)) * (i,1))*> ^ (v1 ^ <*NE*>))) by FINSEQ_1: 32

          .= (( LSeg ((( Gauge (C,n)) * (i,1)),((v1 ^ <*NE*>) /. 1))) \/ ( L~ (v1 ^ <*NE*>))) by SPPOL_2: 20

          .= (( LSeg ((( Gauge (C,n)) * (i,1)),((v1 ^ <*NE*>) /. 1))) \/ (( L~ v1) \/ ( LSeg ((v1 /. ( len v1)),NE)))) by A55, SPPOL_2: 19;

          then

           A108: x in ( LSeg ((( Gauge (C,n)) * (i,1)),((v1 ^ <*NE*>) /. 1))) or x in (( L~ v1) \/ ( LSeg ((v1 /. ( len v1)),NE))) by A106, XBOOLE_0:def 3;

          now

            per cases by A108, XBOOLE_0:def 3;

              suppose x in ( LSeg ((( Gauge (C,n)) * (i,1)),((v1 ^ <*NE*>) /. 1)));

              then x in ( L~ <*(( Gauge (C,n)) * (i,1)), Gij*>) by A58, SPPOL_2: 21;

              hence ( L~ ( Lower_Seq (C,n))) meets ( L~ <*(( Gauge (C,n)) * (i,1)), Gij*>) by A105, A100, XBOOLE_0: 3;

            end;

              suppose

               A109: x in ( L~ v1);

              then x in (( L~ ( Upper_Seq (C,n))) /\ ( L~ ( Lower_Seq (C,n)))) by A105, A100, A86, XBOOLE_0:def 4;

              then x in {( W-min ( L~ ( Cage (C,n)))), ( E-max ( L~ ( Cage (C,n))))} by JORDAN1E: 16;

              then

               A110: x = ( W-min ( L~ ( Cage (C,n)))) by A105, A98, TARSKI:def 2;

              1 in ( dom ( Upper_Seq (C,n))) by A9, FINSEQ_3: 25;

              

              then (( Upper_Seq (C,n)) . 1) = (( Upper_Seq (C,n)) /. 1) by PARTFUN1:def 6

              .= (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) /. 1) by A7, A85, FINSEQ_5: 44

              .= ( W-min ( L~ ( Cage (C,n)))) by A107, FINSEQ_6: 92;

              then x = Gij by A21, A109, A110, JORDAN1E: 7;

              then x in ( LSeg ((( Gauge (C,n)) * (i,1)),Gij)) by RLTOPSP1: 68;

              then x in ( L~ <*(( Gauge (C,n)) * (i,1)), Gij*>) by SPPOL_2: 21;

              hence ( L~ ( Lower_Seq (C,n))) meets ( L~ <*(( Gauge (C,n)) * (i,1)), Gij*>) by A105, A100, XBOOLE_0: 3;

            end;

              suppose

               A111: x in ( LSeg ((v1 /. ( len v1)),NE));

              x in ( L~ ( Cage (C,n))) by A6, A105, A100, XBOOLE_0:def 3;

              then x in (( LSeg (( E-max ( L~ ( Cage (C,n)))),NE)) /\ ( L~ ( Cage (C,n)))) by A102, A111, XBOOLE_0:def 4;

              then x in {( E-max ( L~ ( Cage (C,n))))} by PSCOMP_1: 51;

              hence ( L~ ( Lower_Seq (C,n))) meets ( L~ <*(( Gauge (C,n)) * (i,1)), Gij*>) by A105, A98, TARSKI:def 1;

            end;

          end;

          then ( L~ <*(( Gauge (C,n)) * (i,1)), Gij*>) meets ( L~ ( Lower_Seq (C,n)));

          hence thesis by SPPOL_2: 21;

        end;

          suppose

           A112: Gij in ( L~ ( Upper_Seq (C,n))) & Gij <> (( Upper_Seq (C,n)) . ( len ( Upper_Seq (C,n)))) & ( E-max ( L~ ( Cage (C,n)))) = NE & i > 1;

          now

            let m be Nat;

            assume

             A113: m in ( dom <*(( Gauge (C,n)) * (i,1))*>);

            then m in ( Seg 1) by FINSEQ_1: 38;

            then m = 1 by FINSEQ_1: 2, TARSKI:def 1;

            then ( <*(( Gauge (C,n)) * (i,1))*> . m) = (( Gauge (C,n)) * (i,1)) by FINSEQ_1: 40;

            then

             A114: ( <*(( Gauge (C,n)) * (i,1))*> /. m) = (( Gauge (C,n)) * (i,1)) by A113, PARTFUN1:def 6;

            ( width ( Gauge (C,n))) >= 4 by A12, JORDAN8: 10;

            then

             A115: 1 <= ( width ( Gauge (C,n))) by XXREAL_0: 2;

            then ((( Gauge (C,n)) * (1,1)) `1 ) <= ((( Gauge (C,n)) * (i,1)) `1 ) by A1, A2, SPRECT_3: 13;

            hence ( W-bound ( L~ ( Cage (C,n)))) <= (( <*(( Gauge (C,n)) * (i,1))*> /. m) `1 ) by A12, A114, A115, JORDAN1A: 73;

            ((( Gauge (C,n)) * (i,1)) `1 ) <= ((( Gauge (C,n)) * (( len ( Gauge (C,n))),1)) `1 ) by A1, A2, A115, SPRECT_3: 13;

            hence (( <*(( Gauge (C,n)) * (i,1))*> /. m) `1 ) <= ( E-bound ( L~ ( Cage (C,n)))) by A12, A114, A115, JORDAN1A: 71;

            thus ( S-bound ( L~ ( Cage (C,n)))) <= (( <*(( Gauge (C,n)) * (i,1))*> /. m) `2 ) by A1, A2, A114, JORDAN1A: 72;

            ( S-bound ( L~ ( Cage (C,n)))) = ((( Gauge (C,n)) * (i,1)) `2 ) by A1, A2, JORDAN1A: 72;

            hence (( <*(( Gauge (C,n)) * (i,1))*> /. m) `2 ) <= ( N-bound ( L~ ( Cage (C,n)))) by A114, SPRECT_1: 22;

          end;

          then

           A116: <*(( Gauge (C,n)) * (i,1))*> is_in_the_area_of ( Cage (C,n)) by SPRECT_2:def 1;

           <*Gij*> is_in_the_area_of ( Cage (C,n)) by A112, JORDAN1E: 17, SPRECT_3: 46;

          then v1 is_in_the_area_of ( Cage (C,n)) by A112, JORDAN1E: 17, SPRECT_3: 56;

          then

           A117: Gv1 is_in_the_area_of ( Cage (C,n)) by A116, SPRECT_2: 24;

          

           A118: ( len ( Upper_Seq (C,n))) in ( dom ( Upper_Seq (C,n))) by A9, FINSEQ_3: 25;

          ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 46;

          then

           A119: ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by FINSEQ_6: 90, SPRECT_2: 43;

          

           A120: v1 is non empty by A112, JORDAN1E: 3;

          then

           A121: ( 0 + 1) <= ( len v1) by NAT_1: 13;

          then 1 in ( dom v1) by FINSEQ_3: 25;

          

          then

           A122: (v1 /. 1) = (v1 . 1) by PARTFUN1:def 6

          .= Gij by A112, JORDAN3: 23;

          ( len v1) in ( dom v1) by A121, FINSEQ_3: 25;

          

          then (v1 /. ( len v1)) = (v1 . ( len v1)) by PARTFUN1:def 6

          .= (( Upper_Seq (C,n)) . ( len ( Upper_Seq (C,n)))) by A112, JORDAN1B: 4

          .= (( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) by A118, PARTFUN1:def 6

          .= ((( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) -: ( E-max ( L~ ( Cage (C,n))))) /. (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))))) by A7, A119, FINSEQ_5: 42

          .= ( E-max ( L~ ( Cage (C,n)))) by A119, FINSEQ_5: 45;

          then (Gv1 /. ( len Gv1)) = ( E-max ( L~ ( Cage (C,n)))) by A120, SPRECT_3: 1;

          then

           A123: ((Gv1 /. ( len Gv1)) `2 ) = ( N-bound ( L~ ( Cage (C,n)))) by A112, EUCLID: 52;

          (Gv1 /. 1) = (( Gauge (C,n)) * (i,1)) by FINSEQ_5: 15;

          then ((Gv1 /. 1) `2 ) = ( S-bound ( L~ ( Cage (C,n)))) by A1, A2, JORDAN1A: 72;

          then

           A124: Gv1 is_a_v.c._for ( Cage (C,n)) by A117, A123, SPRECT_2:def 3;

          

           A125: (( Cage (C,n)) /. 1) = ( N-min ( L~ ( Cage (C,n)))) by JORDAN9: 32;

          then (( N-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) <= (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) by SPRECT_2: 70;

          then

           A126: (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) > 1 by A125, SPRECT_2: 69, XXREAL_0: 2;

          (( E-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) <= (( S-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) by A125, SPRECT_2: 72;

          then (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) < (( S-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) by A125, SPRECT_2: 71, XXREAL_0: 2;

          then (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) < (( S-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) by A125, SPRECT_2: 73, XXREAL_0: 2;

          then

           A127: (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) < (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) by A125, SPRECT_2: 74, XXREAL_0: 2;

          then (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) < ( len ( Cage (C,n))) by A125, SPRECT_2: 76, XXREAL_0: 2;

          then

           A128: ((( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) + 1) <= ( len ( Cage (C,n))) by NAT_1: 13;

          

           A129: ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 46;

          then (( Cage (C,n)) /. (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) = ( E-max ( L~ ( Cage (C,n)))) by FINSEQ_5: 38;

          then

           A130: ((( Cage (C,n)) /. ((( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) + 1)) `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by A126, A128, JORDAN1E: 20;

          (1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) <= ( 0 + (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) by A127, NAT_1: 13;

          then ((1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) - (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) <= 0 by XREAL_1: 20;

          then

           A131: (( len ( Cage (C,n))) + ((1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) - (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n))))) <= (( len ( Cage (C,n))) + 0 ) by XREAL_1: 6;

          

           A132: ( len ( Lower_Seq (C,n))) >= (2 + 1) by JORDAN1E: 15;

          then

           A133: ( len ( Lower_Seq (C,n))) > 2 by NAT_1: 13;

          set ci = ( mid (( Upper_Seq (C,n)),(( Index (Gij,( Upper_Seq (C,n)))) + 1),( len ( Upper_Seq (C,n)))));

          ( rng ( Upper_Seq (C,n))) c= ( L~ ( Upper_Seq (C,n))) by A8, SPPOL_2: 18, XXREAL_0: 2;

          then

           A134: not (( Gauge (C,n)) * (i,1)) in ( rng ( Upper_Seq (C,n))) by A2, A112, Th44;

           not (( Gauge (C,n)) * (i,1)) in ( L~ ( Upper_Seq (C,n))) by A2, A112, Th44;

          then not (( Gauge (C,n)) * (i,1)) in {Gij} by A112, TARSKI:def 1;

          then

           A135: not (( Gauge (C,n)) * (i,1)) in ( rng <*Gij*>) by FINSEQ_1: 38;

          now

            per cases ;

              suppose

               A136: Gij <> (( Upper_Seq (C,n)) . (( Index (Gij,( Upper_Seq (C,n)))) + 1));

              ( rng ci) c= ( rng ( Upper_Seq (C,n))) by FINSEQ_6: 119;

              then not (( Gauge (C,n)) * (i,1)) in ( rng ci) by A134;

              then not (( Gauge (C,n)) * (i,1)) in (( rng <*Gij*>) \/ ( rng ci)) by A135, XBOOLE_0:def 3;

              then not (( Gauge (C,n)) * (i,1)) in ( rng ( <*Gij*> ^ ci)) by FINSEQ_1: 31;

              hence not (( Gauge (C,n)) * (i,1)) in ( rng v1) by A136, JORDAN3:def 3;

            end;

              suppose Gij = (( Upper_Seq (C,n)) . (( Index (Gij,( Upper_Seq (C,n)))) + 1));

              then v1 = ci by JORDAN3:def 3;

              then ( rng v1) c= ( rng ( Upper_Seq (C,n))) by FINSEQ_6: 119;

              hence not (( Gauge (C,n)) * (i,1)) in ( rng v1) by A134;

            end;

          end;

          then {(( Gauge (C,n)) * (i,1))} misses ( rng v1) by ZFMISC_1: 50;

          then

           A137: ( rng <*(( Gauge (C,n)) * (i,1))*>) misses ( rng v1) by FINSEQ_1: 38;

          

           A138: (((1 + ((( len ( Cage (C,n))) + (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) - (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n))))) + (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) - ( len ( Cage (C,n)))) = (1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))));

          (1 + ( len v1)) >= (1 + 1) by A121, XREAL_1: 7;

          then

           A139: ( len Gv1) >= 2 by FINSEQ_5: 8;

          3 <= ( len ( Lower_Seq (C,n))) by JORDAN1E: 15;

          then 2 <= ( len ( Lower_Seq (C,n))) by XXREAL_0: 2;

          then

           A140: 2 in ( dom ( Lower_Seq (C,n))) by FINSEQ_3: 25;

          (( Lower_Seq (C,n)) /. 1) = ((( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) :- ( E-max ( L~ ( Cage (C,n))))) /. 1) by JORDAN1E:def 2

          .= ( E-max ( L~ ( Cage (C,n)))) by FINSEQ_5: 53;

          then

           A141: not ( E-max ( L~ ( Cage (C,n)))) in ( L~ h) by A133, JORDAN5B: 16;

          

           A142: v1 is being_S-Seq by A112, JORDAN3: 34;

          (( <*(( Gauge (C,n)) * (i,1))*> /. ( len <*(( Gauge (C,n)) * (i,1))*>)) `1 ) = (( <*(( Gauge (C,n)) * (i,1))*> /. 1) `1 ) by FINSEQ_1: 39

          .= ((( Gauge (C,n)) * (i,1)) `1 ) by FINSEQ_4: 16

          .= ((v1 /. 1) `1 ) by A1, A2, A3, A122, GOBOARD5: 2;

          then

           A143: Gv1 is special by A142, GOBOARD2: 8;

          

           A144: ( L~ Gv1) = (( LSeg ((( Gauge (C,n)) * (i,1)),(v1 /. 1))) \/ ( L~ v1)) by A120, SPPOL_2: 20;

          

           A145: ( W-min ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 43;

          then

           A146: (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) = ((( len ( Cage (C,n))) + (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) - (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) by A129, A127, SPRECT_5: 9;

          (( len ( Cage (C,n))) + 0 ) <= (( len ( Cage (C,n))) + (( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) by XREAL_1: 6;

          then (( len ( Cage (C,n))) - (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) <= (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by A146, XREAL_1: 9;

          then ((( len ( Cage (C,n))) - (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) + 1) <= (1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))))) by XREAL_1: 6;

          then

           A147: ( len (( Cage (C,n)) :- ( W-min ( L~ ( Cage (C,n)))))) <= (1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))))) by A145, FINSEQ_5: 50;

          

           A148: ( len ( Lower_Seq (C,n))) > 1 by A132, XXREAL_0: 2;

          then

           A149: h is non empty by A133, JORDAN1B: 2;

          

           A150: ( len ( Lower_Seq (C,n))) in ( dom ( Lower_Seq (C,n))) by FINSEQ_5: 6;

          then h is_in_the_area_of ( Cage (C,n)) by A140, JORDAN1E: 18, SPRECT_2: 22;

          then

           A151: ( Rev h) is_in_the_area_of ( Cage (C,n)) by SPRECT_3: 51;

          

           A152: ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by A129, FINSEQ_6: 90, SPRECT_2: 43;

          

          then (( Lower_Seq (C,n)) /. (1 + 1)) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) /. (1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))))) by A5, A140, FINSEQ_5: 52

          .= (( Cage (C,n)) /. (((1 + (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))))) + (( W-min ( L~ ( Cage (C,n)))) .. ( Cage (C,n)))) -' ( len ( Cage (C,n))))) by A145, A146, A147, A131, FINSEQ_6: 182

          .= (( Cage (C,n)) /. ((( E-max ( L~ ( Cage (C,n)))) .. ( Cage (C,n))) + 1)) by A146, A138, XREAL_0:def 2;

          then ((h /. 1) `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by A140, A150, A130, SPRECT_2: 8;

          then ((( Rev h) /. ( len h)) `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by A149, FINSEQ_5: 65;

          then

           A153: ((( Rev h) /. ( len ( Rev h))) `1 ) = ( E-bound ( L~ ( Cage (C,n)))) by FINSEQ_5:def 3;

          (( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) /. ( len ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))))) by A5, A152, FINSEQ_5: 54

          .= (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) /. 1) by FINSEQ_6:def 1

          .= ( W-min ( L~ ( Cage (C,n)))) by A145, FINSEQ_6: 92;

          then ((( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

          then ((h /. ( len h)) `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by A140, A150, SPRECT_2: 9;

          then ((( Rev h) /. 1) `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by A149, FINSEQ_5: 65;

          then

           A154: ( Rev h) is_a_h.c._for ( Cage (C,n)) by A151, A153, SPRECT_2:def 2;

           <*(( Gauge (C,n)) * (i,1))*> is one-to-one by FINSEQ_3: 93;

          then

           A155: Gv1 is one-to-one by A137, A142, FINSEQ_3: 91;

          

           A156: ( L~ h) c= ( L~ ( Lower_Seq (C,n))) by A11, JORDAN4: 35;

          h is S-Sequence_in_R2 by A133, A148, JORDAN3: 6;

          then

           A157: ( Rev h) is S-Sequence_in_R2;

          then 2 <= ( len ( Rev h)) by TOPREAL1:def 8;

          then ( L~ ( Rev h)) meets ( L~ Gv1) by A139, A155, A143, A157, A154, A124, SPRECT_2: 29;

          then ( L~ h) meets ( L~ Gv1) by SPPOL_2: 22;

          then

          consider x be object such that

           A158: x in ( L~ h) and

           A159: x in ( L~ Gv1) by XBOOLE_0: 3;

          

           A160: ( W-min ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 43;

          

           A161: ( L~ v1) c= ( L~ ( Upper_Seq (C,n))) by A112, JORDAN3: 42;

          now

            per cases by A159, A144, XBOOLE_0:def 3;

              suppose x in ( LSeg ((( Gauge (C,n)) * (i,1)),(v1 /. 1)));

              then x in ( L~ <*(( Gauge (C,n)) * (i,1)), Gij*>) by A122, SPPOL_2: 21;

              hence ( L~ ( Lower_Seq (C,n))) meets ( L~ <*(( Gauge (C,n)) * (i,1)), Gij*>) by A158, A156, XBOOLE_0: 3;

            end;

              suppose

               A162: x in ( L~ v1);

              then x in (( L~ ( Upper_Seq (C,n))) /\ ( L~ ( Lower_Seq (C,n)))) by A158, A156, A161, XBOOLE_0:def 4;

              then x in {( W-min ( L~ ( Cage (C,n)))), ( E-max ( L~ ( Cage (C,n))))} by JORDAN1E: 16;

              then

               A163: x = ( W-min ( L~ ( Cage (C,n)))) by A158, A141, TARSKI:def 2;

              1 in ( dom ( Upper_Seq (C,n))) by A9, FINSEQ_3: 25;

              

              then (( Upper_Seq (C,n)) . 1) = (( Upper_Seq (C,n)) /. 1) by PARTFUN1:def 6

              .= (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) /. 1) by A7, A119, FINSEQ_5: 44

              .= ( W-min ( L~ ( Cage (C,n)))) by A160, FINSEQ_6: 92;

              then x = Gij by A112, A162, A163, JORDAN1E: 7;

              then x in ( LSeg ((( Gauge (C,n)) * (i,1)),Gij)) by RLTOPSP1: 68;

              then x in ( L~ <*(( Gauge (C,n)) * (i,1)), Gij*>) by SPPOL_2: 21;

              hence ( L~ ( Lower_Seq (C,n))) meets ( L~ <*(( Gauge (C,n)) * (i,1)), Gij*>) by A158, A156, XBOOLE_0: 3;

            end;

          end;

          then ( L~ <*(( Gauge (C,n)) * (i,1)), Gij*>) meets ( L~ ( Lower_Seq (C,n)));

          hence thesis by SPPOL_2: 21;

        end;

          suppose

           A164: Gij in ( L~ ( Lower_Seq (C,n)));

          Gij in ( LSeg ((( Gauge (C,n)) * (i,1)),Gij)) by RLTOPSP1: 68;

          hence thesis by A164, XBOOLE_0: 3;

        end;

          suppose

           A165: Gij in ( L~ ( Upper_Seq (C,n))) & Gij = (( Upper_Seq (C,n)) . ( len ( Upper_Seq (C,n))));

          

           A166: Gij in ( LSeg ((( Gauge (C,n)) * (i,1)),Gij)) by RLTOPSP1: 68;

          

           A167: ( rng ( Lower_Seq (C,n))) c= ( L~ ( Lower_Seq (C,n))) & ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Lower_Seq (C,n))) by A5, A10, FINSEQ_6: 61, SPPOL_2: 18, XXREAL_0: 2;

          ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 46;

          then

           A168: ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by FINSEQ_6: 90, SPRECT_2: 43;

          ( len ( Upper_Seq (C,n))) in ( dom ( Upper_Seq (C,n))) by A9, FINSEQ_3: 25;

          

          then (( Upper_Seq (C,n)) . ( len ( Upper_Seq (C,n)))) = (( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) by PARTFUN1:def 6

          .= ((( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) -: ( E-max ( L~ ( Cage (C,n))))) /. (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))))) by A7, A168, FINSEQ_5: 42

          .= ( E-max ( L~ ( Cage (C,n)))) by A168, FINSEQ_5: 45;

          hence thesis by A165, A167, A166, XBOOLE_0: 3;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN1G:47

    

     Th47: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for n be Nat st n > 0 holds ( First_Point (( L~ ( Upper_Seq (C,n))),( W-min ( L~ ( Cage (C,n)))),( E-max ( L~ ( Cage (C,n)))),( Vertical_Line ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)))) in ( rng ( Upper_Seq (C,n)))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let n be Nat;

      assume

       A1: n > 0 ;

      set sr = ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2);

      set Ebo = ( E-bound ( L~ ( Cage (C,n))));

      set Wbo = ( W-bound ( L~ ( Cage (C,n))));

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      set FiP = ( First_Point (( L~ ( Upper_Seq (C,n))),Wmin,Emax,( Vertical_Line sr)));

      

       A2: 1 <= ( Center ( Gauge (C,n))) by JORDAN1B: 11;

      

       A3: (( Upper_Seq (C,n)) /. 1) = ( W-min ( L~ ( Cage (C,n)))) & (( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) = ( E-max ( L~ ( Cage (C,n)))) by JORDAN1F: 5, JORDAN1F: 7;

      then

       A4: ( L~ ( Upper_Seq (C,n))) is_an_arc_of (Wmin,Emax) by TOPREAL1: 25;

      

       A5: Wbo < Ebo by SPRECT_1: 31;

      then Wbo < sr by XREAL_1: 226;

      then

       A6: (Wmin `1 ) <= sr by EUCLID: 52;

      

       A7: ( Center ( Gauge (C,n))) <= ( len ( Gauge (C,n))) by JORDAN1B: 13;

      sr < Ebo by A5, XREAL_1: 226;

      then

       A8: sr <= (Emax `1 ) by EUCLID: 52;

      then

       A9: ( L~ ( Upper_Seq (C,n))) meets ( Vertical_Line sr) by A4, A6, JORDAN6: 49;

      (( L~ ( Upper_Seq (C,n))) /\ ( Vertical_Line sr)) is closed by A4, A6, A8, JORDAN6: 49;

      then

       A10: FiP in (( L~ ( Upper_Seq (C,n))) /\ ( Vertical_Line sr)) by A4, A9, JORDAN5C:def 1;

      then FiP in ( L~ ( Upper_Seq (C,n))) by XBOOLE_0:def 4;

      then

      consider t be Nat such that

       A11: 1 <= t and

       A12: (t + 1) <= ( len ( Upper_Seq (C,n))) and

       A13: FiP in ( LSeg (( Upper_Seq (C,n)),t)) by SPPOL_2: 13;

      

       A14: ( LSeg (( Upper_Seq (C,n)),t)) = ( LSeg ((( Upper_Seq (C,n)) /. t),(( Upper_Seq (C,n)) /. (t + 1)))) by A11, A12, TOPREAL1:def 3;

      t < ( len ( Upper_Seq (C,n))) by A12, NAT_1: 13;

      then

       A15: t in ( dom ( Upper_Seq (C,n))) by A11, FINSEQ_3: 25;

      1 <= (t + 1) by A11, NAT_1: 13;

      then

       A16: (t + 1) in ( dom ( Upper_Seq (C,n))) by A12, FINSEQ_3: 25;

      FiP in ( Vertical_Line sr) by A10, XBOOLE_0:def 4;

      then

       A17: (FiP `1 ) = sr by JORDAN6: 31;

      

       A18: FiP = ( First_Point (( LSeg (( Upper_Seq (C,n)),t)),(( Upper_Seq (C,n)) /. t),(( Upper_Seq (C,n)) /. (t + 1)),( Vertical_Line sr))) by A3, A9, A11, A12, A13, JORDAN5C: 19, JORDAN6: 30;

      now

        per cases by SPPOL_1: 19;

          suppose

           A19: ( LSeg (( Upper_Seq (C,n)),t)) is vertical;

          then ((( Upper_Seq (C,n)) /. (t + 1)) `1 ) = sr by A13, A14, A17, SPPOL_1: 41;

          then (( Upper_Seq (C,n)) /. (t + 1)) in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = sr };

          then

           A20: (( Upper_Seq (C,n)) /. (t + 1)) in ( Vertical_Line sr) by JORDAN6:def 6;

          

           A21: ( LSeg (( Upper_Seq (C,n)),t)) is closed & ( LSeg (( Upper_Seq (C,n)),t)) is_an_arc_of ((( Upper_Seq (C,n)) /. t),(( Upper_Seq (C,n)) /. (t + 1))) by A14, A15, A16, GOBOARD7: 29, TOPREAL1: 9;

          ((( Upper_Seq (C,n)) /. t) `1 ) = sr by A13, A14, A17, A19, SPPOL_1: 41;

          then (( Upper_Seq (C,n)) /. t) in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = sr };

          then (( Upper_Seq (C,n)) /. t) in ( Vertical_Line sr) by JORDAN6:def 6;

          then ( LSeg (( Upper_Seq (C,n)),t)) c= ( Vertical_Line sr) by A14, A20, JORDAN1A: 13;

          then ( First_Point (( LSeg (( Upper_Seq (C,n)),t)),(( Upper_Seq (C,n)) /. t),(( Upper_Seq (C,n)) /. (t + 1)),( Vertical_Line sr))) = (( Upper_Seq (C,n)) /. t) by A21, JORDAN5C: 7;

          hence thesis by A18, A15, PARTFUN2: 2;

        end;

          suppose ( LSeg (( Upper_Seq (C,n)),t)) is horizontal;

          then

           A22: ((( Upper_Seq (C,n)) /. t) `2 ) = ((( Upper_Seq (C,n)) /. (t + 1)) `2 ) by A14, SPPOL_1: 15;

          then

           A23: (FiP `2 ) = ((( Upper_Seq (C,n)) /. t) `2 ) by A13, A14, GOBOARD7: 6;

          ( Upper_Seq (C,n)) is_sequence_on ( Gauge (C,n)) by Th4;

          then

          consider i1,j1,i2,j2 be Nat such that

           A24: [i1, j1] in ( Indices ( Gauge (C,n))) and

           A25: (( Upper_Seq (C,n)) /. t) = (( Gauge (C,n)) * (i1,j1)) and

           A26: [i2, j2] in ( Indices ( Gauge (C,n))) and

           A27: (( Upper_Seq (C,n)) /. (t + 1)) = (( Gauge (C,n)) * (i2,j2)) and

           A28: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A11, A12, JORDAN8: 3;

          

           A29: 1 <= i1 by A24, MATRIX_0: 32;

          

           A30: 1 <= i2 by A26, MATRIX_0: 32;

          

           A31: i1 <= ( len ( Gauge (C,n))) by A24, MATRIX_0: 32;

          

           A32: j1 = j2 by A22, A24, A25, A26, A27, Th6;

          

           A33: i2 <= ( len ( Gauge (C,n))) by A26, MATRIX_0: 32;

          

           A34: 1 <= j1 & j1 <= ( width ( Gauge (C,n))) by A24, MATRIX_0: 32;

          

          then

           A35: ((( Gauge (C,n)) * (( Center ( Gauge (C,n))),j1)) `1 ) = ((( W-bound C) + ( E-bound C)) / 2) by A1, Th35

          .= (FiP `1 ) by A17, Th33;

          ((( Gauge (C,n)) * (( Center ( Gauge (C,n))),j1)) `2 ) = ((( Gauge (C,n)) * (1,j1)) `2 ) by A2, A7, A34, GOBOARD5: 1

          .= (FiP `2 ) by A23, A25, A29, A31, A34, GOBOARD5: 1;

          then

           A36: FiP = (( Gauge (C,n)) * (( Center ( Gauge (C,n))),j1)) by A35, TOPREAL3: 6;

          now

            per cases by A28, A32;

              suppose

               A37: (i1 + 1) = i2;

              i1 < (i1 + 1) by NAT_1: 13;

              then

               A38: ((( Gauge (C,n)) * (i1,j1)) `1 ) <= ((( Gauge (C,n)) * ((i1 + 1),j1)) `1 ) by A29, A34, A33, A37, SPRECT_3: 13;

              then ((( Gauge (C,n)) * (i1,j1)) `1 ) <= (FiP `1 ) by A13, A14, A25, A27, A32, A37, TOPREAL1: 3;

              then i1 <= ( Center ( Gauge (C,n))) by A2, A31, A34, A35, GOBOARD5: 3;

              then i1 = ( Center ( Gauge (C,n))) or i1 < ( Center ( Gauge (C,n))) by XXREAL_0: 1;

              then

               A39: i1 = ( Center ( Gauge (C,n))) or (i1 + 1) <= ( Center ( Gauge (C,n))) by NAT_1: 13;

              (FiP `1 ) <= ((( Gauge (C,n)) * ((i1 + 1),j1)) `1 ) by A13, A14, A25, A27, A32, A37, A38, TOPREAL1: 3;

              then ( Center ( Gauge (C,n))) <= (i1 + 1) by A7, A34, A30, A35, A37, GOBOARD5: 3;

              then i1 = ( Center ( Gauge (C,n))) or (i1 + 1) = ( Center ( Gauge (C,n))) by A39, XXREAL_0: 1;

              hence thesis by A15, A16, A25, A27, A32, A36, A37, PARTFUN2: 2;

            end;

              suppose

               A40: i1 = (i2 + 1);

              i2 < (i2 + 1) by NAT_1: 13;

              then

               A41: ((( Gauge (C,n)) * (i2,j1)) `1 ) <= ((( Gauge (C,n)) * ((i2 + 1),j1)) `1 ) by A31, A34, A30, A40, SPRECT_3: 13;

              then ((( Gauge (C,n)) * (i2,j1)) `1 ) <= (FiP `1 ) by A13, A14, A25, A27, A32, A40, TOPREAL1: 3;

              then i2 <= ( Center ( Gauge (C,n))) by A2, A34, A33, A35, GOBOARD5: 3;

              then i2 = ( Center ( Gauge (C,n))) or i2 < ( Center ( Gauge (C,n))) by XXREAL_0: 1;

              then

               A42: i2 = ( Center ( Gauge (C,n))) or (i2 + 1) <= ( Center ( Gauge (C,n))) by NAT_1: 13;

              (FiP `1 ) <= ((( Gauge (C,n)) * ((i2 + 1),j1)) `1 ) by A13, A14, A25, A27, A32, A40, A41, TOPREAL1: 3;

              then ( Center ( Gauge (C,n))) <= (i2 + 1) by A7, A29, A34, A35, A40, GOBOARD5: 3;

              then i2 = ( Center ( Gauge (C,n))) or (i2 + 1) = ( Center ( Gauge (C,n))) by A42, XXREAL_0: 1;

              hence thesis by A15, A16, A25, A27, A32, A36, A40, PARTFUN2: 2;

            end;

          end;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN1G:48

    

     Th48: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for n be Nat st n > 0 holds ( Last_Point (( L~ ( Lower_Seq (C,n))),( E-max ( L~ ( Cage (C,n)))),( W-min ( L~ ( Cage (C,n)))),( Vertical_Line ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)))) in ( rng ( Lower_Seq (C,n)))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let n be Nat;

      assume

       A1: n > 0 ;

      set sr = ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2);

      set Ebo = ( E-bound ( L~ ( Cage (C,n))));

      set Wbo = ( W-bound ( L~ ( Cage (C,n))));

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      set LaP = ( Last_Point (( L~ ( Lower_Seq (C,n))),Emax,Wmin,( Vertical_Line sr)));

      

       A2: (( Lower_Seq (C,n)) /. 1) = ( E-max ( L~ ( Cage (C,n)))) & (( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) = ( W-min ( L~ ( Cage (C,n)))) by JORDAN1F: 6, JORDAN1F: 8;

      then

       A3: ( L~ ( Lower_Seq (C,n))) is_an_arc_of (Emax,Wmin) by TOPREAL1: 25;

      

       A4: Wbo <= Ebo by SPRECT_1: 21;

      then Wbo <= sr by JORDAN6: 1;

      then

       A5: (Wmin `1 ) <= sr by EUCLID: 52;

      sr <= Ebo by A4, JORDAN6: 1;

      then

       A6: sr <= (Emax `1 ) by EUCLID: 52;

      

       A7: ( L~ ( Lower_Seq (C,n))) is_an_arc_of (Wmin,Emax) by A2, JORDAN5B: 14, TOPREAL1: 25;

      then

       A8: ( L~ ( Lower_Seq (C,n))) meets ( Vertical_Line sr) by A5, A6, JORDAN6: 49;

      (( L~ ( Lower_Seq (C,n))) /\ ( Vertical_Line sr)) is closed by A7, A5, A6, JORDAN6: 49;

      then

       A9: LaP in (( L~ ( Lower_Seq (C,n))) /\ ( Vertical_Line sr)) by A3, A8, JORDAN5C:def 2;

      then LaP in ( L~ ( Lower_Seq (C,n))) by XBOOLE_0:def 4;

      then

      consider t be Nat such that

       A10: 1 <= t and

       A11: (t + 1) <= ( len ( Lower_Seq (C,n))) and

       A12: LaP in ( LSeg (( Lower_Seq (C,n)),t)) by SPPOL_2: 13;

      

       A13: ( LSeg (( Lower_Seq (C,n)),t)) = ( LSeg ((( Lower_Seq (C,n)) /. t),(( Lower_Seq (C,n)) /. (t + 1)))) by A10, A11, TOPREAL1:def 3;

      1 <= (t + 1) by A10, NAT_1: 13;

      then

       A14: (t + 1) in ( dom ( Lower_Seq (C,n))) by A11, FINSEQ_3: 25;

      t < ( len ( Lower_Seq (C,n))) by A11, NAT_1: 13;

      then

       A15: t in ( dom ( Lower_Seq (C,n))) by A10, FINSEQ_3: 25;

      LaP in ( Vertical_Line sr) by A9, XBOOLE_0:def 4;

      then

       A16: (LaP `1 ) = sr by JORDAN6: 31;

      

       A17: LaP = ( Last_Point (( LSeg (( Lower_Seq (C,n)),t)),(( Lower_Seq (C,n)) /. t),(( Lower_Seq (C,n)) /. (t + 1)),( Vertical_Line sr))) by A2, A8, A10, A11, A12, JORDAN5C: 20, JORDAN6: 30;

      now

        per cases by SPPOL_1: 19;

          suppose

           A18: ( LSeg (( Lower_Seq (C,n)),t)) is vertical;

          then ((( Lower_Seq (C,n)) /. (t + 1)) `1 ) = sr by A12, A13, A16, SPPOL_1: 41;

          then (( Lower_Seq (C,n)) /. (t + 1)) in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = sr };

          then

           A19: (( Lower_Seq (C,n)) /. (t + 1)) in ( Vertical_Line sr) by JORDAN6:def 6;

          

           A20: ( LSeg (( Lower_Seq (C,n)),t)) is closed & ( LSeg (( Lower_Seq (C,n)),t)) is_an_arc_of ((( Lower_Seq (C,n)) /. t),(( Lower_Seq (C,n)) /. (t + 1))) by A13, A15, A14, GOBOARD7: 29, TOPREAL1: 9;

          ((( Lower_Seq (C,n)) /. t) `1 ) = sr by A12, A13, A16, A18, SPPOL_1: 41;

          then (( Lower_Seq (C,n)) /. t) in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = sr };

          then (( Lower_Seq (C,n)) /. t) in ( Vertical_Line sr) by JORDAN6:def 6;

          then ( LSeg (( Lower_Seq (C,n)),t)) c= ( Vertical_Line sr) by A13, A19, JORDAN1A: 13;

          then ( Last_Point (( LSeg (( Lower_Seq (C,n)),t)),(( Lower_Seq (C,n)) /. t),(( Lower_Seq (C,n)) /. (t + 1)),( Vertical_Line sr))) = (( Lower_Seq (C,n)) /. (t + 1)) by A20, JORDAN5C: 7;

          hence thesis by A17, A14, PARTFUN2: 2;

        end;

          suppose ( LSeg (( Lower_Seq (C,n)),t)) is horizontal;

          then

           A21: ((( Lower_Seq (C,n)) /. t) `2 ) = ((( Lower_Seq (C,n)) /. (t + 1)) `2 ) by A13, SPPOL_1: 15;

          then

           A22: (LaP `2 ) = ((( Lower_Seq (C,n)) /. t) `2 ) by A12, A13, GOBOARD7: 6;

          ( Lower_Seq (C,n)) is_sequence_on ( Gauge (C,n)) by Th5;

          then

          consider i1,j1,i2,j2 be Nat such that

           A23: [i1, j1] in ( Indices ( Gauge (C,n))) and

           A24: (( Lower_Seq (C,n)) /. t) = (( Gauge (C,n)) * (i1,j1)) and

           A25: [i2, j2] in ( Indices ( Gauge (C,n))) and

           A26: (( Lower_Seq (C,n)) /. (t + 1)) = (( Gauge (C,n)) * (i2,j2)) and

           A27: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A10, A11, JORDAN8: 3;

          

           A28: 1 <= i1 by A23, MATRIX_0: 32;

          

           A29: j1 = j2 by A21, A23, A24, A25, A26, Th6;

          

           A30: i2 <= ( len ( Gauge (C,n))) by A25, MATRIX_0: 32;

          

           A31: i1 <= ( len ( Gauge (C,n))) by A23, MATRIX_0: 32;

          

           A32: 1 <= i2 by A25, MATRIX_0: 32;

          

           A33: ( Center ( Gauge (C,n))) <= ( len ( Gauge (C,n))) by JORDAN1B: 13;

          

           A34: 1 <= j1 & j1 <= ( width ( Gauge (C,n))) by A23, MATRIX_0: 32;

          

          then

           A35: ((( Gauge (C,n)) * (( Center ( Gauge (C,n))),j1)) `1 ) = ((( W-bound C) + ( E-bound C)) / 2) by A1, Th35

          .= (LaP `1 ) by A16, Th33;

          

           A36: 1 <= ( Center ( Gauge (C,n))) by JORDAN1B: 11;

          

          then ((( Gauge (C,n)) * (( Center ( Gauge (C,n))),j1)) `2 ) = ((( Gauge (C,n)) * (1,j1)) `2 ) by A34, A33, GOBOARD5: 1

          .= (LaP `2 ) by A22, A24, A28, A31, A34, GOBOARD5: 1;

          then

           A37: LaP = (( Gauge (C,n)) * (( Center ( Gauge (C,n))),j1)) by A35, TOPREAL3: 6;

          now

            per cases by A27, A29;

              suppose

               A38: (i1 + 1) = i2;

              i1 < (i1 + 1) by NAT_1: 13;

              then

               A39: ((( Gauge (C,n)) * (i1,j1)) `1 ) <= ((( Gauge (C,n)) * ((i1 + 1),j1)) `1 ) by A28, A34, A30, A38, SPRECT_3: 13;

              then ((( Gauge (C,n)) * (i1,j1)) `1 ) <= (LaP `1 ) by A12, A13, A24, A26, A29, A38, TOPREAL1: 3;

              then i1 <= ( Center ( Gauge (C,n))) by A31, A34, A36, A35, GOBOARD5: 3;

              then i1 = ( Center ( Gauge (C,n))) or i1 < ( Center ( Gauge (C,n))) by XXREAL_0: 1;

              then

               A40: i1 = ( Center ( Gauge (C,n))) or (i1 + 1) <= ( Center ( Gauge (C,n))) by NAT_1: 13;

              (LaP `1 ) <= ((( Gauge (C,n)) * ((i1 + 1),j1)) `1 ) by A12, A13, A24, A26, A29, A38, A39, TOPREAL1: 3;

              then ( Center ( Gauge (C,n))) <= (i1 + 1) by A34, A32, A33, A35, A38, GOBOARD5: 3;

              then i1 = ( Center ( Gauge (C,n))) or (i1 + 1) = ( Center ( Gauge (C,n))) by A40, XXREAL_0: 1;

              hence thesis by A15, A14, A24, A26, A29, A37, A38, PARTFUN2: 2;

            end;

              suppose

               A41: i1 = (i2 + 1);

              i2 < (i2 + 1) by NAT_1: 13;

              then

               A42: ((( Gauge (C,n)) * (i2,j1)) `1 ) <= ((( Gauge (C,n)) * ((i2 + 1),j1)) `1 ) by A31, A34, A32, A41, SPRECT_3: 13;

              then ((( Gauge (C,n)) * (i2,j1)) `1 ) <= (LaP `1 ) by A12, A13, A24, A26, A29, A41, TOPREAL1: 3;

              then i2 <= ( Center ( Gauge (C,n))) by A34, A30, A36, A35, GOBOARD5: 3;

              then i2 = ( Center ( Gauge (C,n))) or i2 < ( Center ( Gauge (C,n))) by XXREAL_0: 1;

              then

               A43: i2 = ( Center ( Gauge (C,n))) or (i2 + 1) <= ( Center ( Gauge (C,n))) by NAT_1: 13;

              (LaP `1 ) <= ((( Gauge (C,n)) * ((i2 + 1),j1)) `1 ) by A12, A13, A24, A26, A29, A41, A42, TOPREAL1: 3;

              then ( Center ( Gauge (C,n))) <= (i2 + 1) by A28, A34, A33, A35, A41, GOBOARD5: 3;

              then i2 = ( Center ( Gauge (C,n))) or (i2 + 1) = ( Center ( Gauge (C,n))) by A43, XXREAL_0: 1;

              hence thesis by A15, A14, A24, A26, A29, A37, A41, PARTFUN2: 2;

            end;

          end;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: JORDAN1G:49

    

     Th49: for f be S-Sequence_in_R2 holds for p be Point of ( TOP-REAL 2) st p in ( rng f) holds ( R_Cut (f,p)) = ( mid (f,1,(p .. f)))

    proof

      let f be S-Sequence_in_R2;

      let p be Point of ( TOP-REAL 2);

      assume

       A1: p in ( rng f);

      then

      consider i be Nat such that

       A2: i in ( dom f) and

       A3: (f . i) = p by FINSEQ_2: 10;

      reconsider i as Nat;

      

       A4: i <= ( len f) by A2, FINSEQ_3: 25;

      ( len f) >= 2 by TOPREAL1:def 8;

      then

       A5: ( rng f) c= ( L~ f) by SPPOL_2: 18;

      then

       A6: 1 <= ( Index (p,f)) by A1, JORDAN3: 8;

      

       A7: ( Index (p,f)) < ( len f) by A1, A5, JORDAN3: 8;

      

       A8: ( 0 + 1) <= i by A2, FINSEQ_3: 25;

      then

       A9: (i - 1) >= 0 by XREAL_1: 19;

      per cases by A8, XXREAL_0: 1;

        suppose

         A10: 1 < i;

        1 <= ( len f) by A8, A4, XXREAL_0: 2;

        then 1 in ( dom f) by FINSEQ_3: 25;

        then p <> (f . 1) by A2, A3, A10, FUNCT_1:def 4;

        then

         A11: ( R_Cut (f,p)) = (( mid (f,1,( Index (p,f)))) ^ <*p*>) by JORDAN3:def 4;

        

         A12: (( Index (p,f)) + 1) = i by A3, A4, A10, JORDAN3: 12;

        

         A13: ( len ( mid (f,1,( Index (p,f))))) = ((( Index (p,f)) -' 1) + 1) by A6, A7, JORDAN4: 8

        .= (i -' 1) by A1, A5, A12, JORDAN3: 8, NAT_D: 38;

        

         A14: ( len ( mid (f,1,i))) = ((i -' 1) + 1) by A8, A4, JORDAN4: 8

        .= i by A8, XREAL_1: 235;

        then

         A15: ( dom ( mid (f,1,i))) = ( Seg i) by FINSEQ_1:def 3;

         A16:

        now

          let j be Nat;

          reconsider a = j as Nat;

          assume

           A17: j in ( dom ( mid (f,1,i)));

          then

           A18: 1 <= j by A15, FINSEQ_1: 1;

          

           A19: j <= i by A15, A17, FINSEQ_1: 1;

          now

            per cases by A19, XXREAL_0: 1;

              suppose j < i;

              then

               A20: j <= ( Index (p,f)) by A12, NAT_1: 13;

              then j <= (i -' 1) by A9, A12, XREAL_0:def 2;

              then

               A21: j in ( dom ( mid (f,1,( Index (p,f))))) by A13, A18, FINSEQ_3: 25;

              

              thus (( mid (f,1,i)) . j) = (f . a) by A4, A18, A19, FINSEQ_6: 123

              .= (( mid (f,1,( Index (p,f)))) . a) by A7, A18, A20, FINSEQ_6: 123

              .= ((( mid (f,1,( Index (p,f)))) ^ <*p*>) . j) by A21, FINSEQ_1:def 7;

            end;

              suppose

               A22: j = i;

              

               A23: ((i -' 1) + 1) = i by A8, XREAL_1: 235;

              

              thus (( mid (f,1,i)) . j) = (f . a) by A4, A18, A19, FINSEQ_6: 123

              .= ((( mid (f,1,( Index (p,f)))) ^ <*p*>) . j) by A3, A13, A22, A23, FINSEQ_1: 42;

            end;

          end;

          hence (( mid (f,1,i)) . j) = ((( mid (f,1,( Index (p,f)))) ^ <*p*>) . j);

        end;

        ( len (( mid (f,1,( Index (p,f)))) ^ <*p*>)) = ((i -' 1) + 1) by A13, FINSEQ_2: 16

        .= i by A8, XREAL_1: 235;

        then ( mid (f,1,i)) = ( R_Cut (f,p)) by A11, A14, A16, FINSEQ_2: 9;

        hence thesis by A2, A3, FINSEQ_5: 11;

      end;

        suppose

         A24: 1 = i;

        then

         A25: ( R_Cut (f,p)) = <*p*> by A3, JORDAN3:def 4;

        

         A26: p = (f /. 1) by A2, A3, A24, PARTFUN1:def 6;

        then

         S: (p .. f) = 1 by FINSEQ_6: 43;

        (f /. 1) = (f . 1) by A2, PARTFUN1:def 6, A24;

        hence thesis by A2, A24, A25, A26, S, JORDAN4: 15;

      end;

    end;

    theorem :: JORDAN1G:50

    

     Th50: for f be S-Sequence_in_R2 holds for Q be closed Subset of ( TOP-REAL 2) st ( L~ f) meets Q & not (f /. 1) in Q & ( First_Point (( L~ f),(f /. 1),(f /. ( len f)),Q)) in ( rng f) holds (( L~ ( mid (f,1,(( First_Point (( L~ f),(f /. 1),(f /. ( len f)),Q)) .. f)))) /\ Q) = {( First_Point (( L~ f),(f /. 1),(f /. ( len f)),Q))}

    proof

      let f be S-Sequence_in_R2;

      let Q be closed Subset of ( TOP-REAL 2);

      assume that

       A1: ( L~ f) meets Q & not (f /. 1) in Q and

       A2: ( First_Point (( L~ f),(f /. 1),(f /. ( len f)),Q)) in ( rng f);

      (( L~ ( R_Cut (f,( First_Point (( L~ f),(f /. 1),(f /. ( len f)),Q))))) /\ Q) = {( First_Point (( L~ f),(f /. 1),(f /. ( len f)),Q))} by A1, SPRECT_4: 1;

      hence thesis by A2, Th49;

    end;

    theorem :: JORDAN1G:51

    

     Th51: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for n be Nat st n > 0 holds for k be Nat st 1 <= k & k < (( First_Point (( L~ ( Upper_Seq (C,n))),( W-min ( L~ ( Cage (C,n)))),( E-max ( L~ ( Cage (C,n)))),( Vertical_Line ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)))) .. ( Upper_Seq (C,n))) holds ((( Upper_Seq (C,n)) /. k) `1 ) < ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let n be Nat;

      assume

       A1: n > 0 ;

      set US = ( Upper_Seq (C,n));

      set sr = ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2);

      set Ebo = ( E-bound ( L~ ( Cage (C,n))));

      set Wbo = ( W-bound ( L~ ( Cage (C,n))));

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      set FiP = ( First_Point (( L~ US),Wmin,Emax,( Vertical_Line sr)));

      defpred P[ Nat] means 1 <= $1 & $1 < (FiP .. US) implies ((US /. $1) `1 ) < sr;

      

       A2: Wbo < Ebo by SPRECT_1: 31;

      then

       A3: Wbo < sr by XREAL_1: 226;

      

       A4: sr < Ebo by A2, XREAL_1: 226;

      

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

      proof

        set GC1 = (( Gauge (C,n)) * (( Center ( Gauge (C,n))),1));

        let k be non zero Nat;

        assume

         A6: 1 <= k & k < (FiP .. US) implies ((US /. k) `1 ) < sr;

        4 <= ( len ( Gauge (C,n))) by JORDAN8: 10;

        then 1 <= ( len ( Gauge (C,n))) by XXREAL_0: 2;

        then

         A7: 1 <= ( width ( Gauge (C,n))) by JORDAN8:def 1;

        

        then

         A8: (GC1 `1 ) = ((( W-bound C) + ( E-bound C)) / 2) by A1, Th35

        .= sr by Th33;

        

         A9: k >= 1 by NAT_1: 14;

        

         A10: (US /. ( len US)) = Emax by JORDAN1F: 7;

        

         A11: FiP in ( rng US) by A1, Th47;

        then

         A12: (FiP .. ( Upper_Seq (C,n))) in ( dom ( Upper_Seq (C,n))) by FINSEQ_4: 20;

        then

         A13: 1 <= (FiP .. US) by FINSEQ_3: 25;

        

         A14: 1 <= ( Center ( Gauge (C,n))) by JORDAN1B: 11;

        

         A15: (US /. 1) = Wmin by JORDAN1F: 5;

        reconsider kk = k as Nat;

        assume that

         A16: 1 <= (k + 1) and

         A17: (k + 1) < (FiP .. US);

        

         A18: (FiP .. US) <= ( len US) by A12, FINSEQ_3: 25;

        then

         A19: (k + 1) <= ( len US) by A17, XXREAL_0: 2;

        US is_sequence_on ( Gauge (C,n)) by Th4;

        then

        consider i1,j1,i2,j2 be Nat such that

         A20: [i1, j1] in ( Indices ( Gauge (C,n))) and

         A21: (US /. kk) = (( Gauge (C,n)) * (i1,j1)) and

         A22: [i2, j2] in ( Indices ( Gauge (C,n))) and

         A23: (US /. (kk + 1)) = (( Gauge (C,n)) * (i2,j2)) and

         A24: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A9, A19, JORDAN8: 3;

        

         A25: 1 <= i1 by A20, MATRIX_0: 32;

        

         A26: 1 <= j1 & j1 <= ( width ( Gauge (C,n))) by A20, MATRIX_0: 32;

        

         A27: j2 <= ( width ( Gauge (C,n))) by A22, MATRIX_0: 32;

        

         A28: 1 <= i2 & 1 <= j2 by A22, MATRIX_0: 32;

        

         A29: i2 <= ( len ( Gauge (C,n))) by A22, MATRIX_0: 32;

        

         A30: i1 <= ( len ( Gauge (C,n))) by A20, MATRIX_0: 32;

        

         A31: ( Center ( Gauge (C,n))) <= ( len ( Gauge (C,n))) & (i1 + 1) >= 1 by JORDAN1B: 13, NAT_1: 11;

        now

          per cases by A24;

            suppose i1 = i2 & (j1 + 1) = j2;

            

            then ((US /. k) `1 ) = ((( Gauge (C,n)) * (i2,1)) `1 ) by A21, A25, A30, A26, GOBOARD5: 2

            .= ((US /. (k + 1)) `1 ) by A23, A29, A28, A27, GOBOARD5: 2;

            hence thesis by A6, A17, NAT_1: 13, NAT_1: 14;

          end;

            suppose

             A32: (i1 + 1) = i2 & j1 = j2;

             A33:

            now

              

               A34: (k + 1) >= (1 + 1) by A9, XREAL_1: 7;

              ( len ( mid (US,1,(FiP .. US)))) = (((FiP .. US) -' 1) + 1) by A13, A18, JORDAN4: 8

              .= (FiP .. US) by A13, XREAL_1: 235;

              then

               A35: ( rng ( mid (US,1,(FiP .. US)))) c= ( L~ ( mid (US,1,(FiP .. US)))) by A17, A34, SPPOL_2: 18, XXREAL_0: 2;

              

               A36: (US /. (FiP .. US)) = FiP by A11, FINSEQ_5: 38;

               A37:

              now

                assume (US /. 1) in ( Vertical_Line sr);

                then (Wmin `1 ) = sr by A15, JORDAN6: 31;

                hence contradiction by A3, EUCLID: 52;

              end;

              

               A38: (Wmin `1 ) <= sr & sr <= (Emax `1 ) by A3, A4, EUCLID: 52;

              

               A39: ( Vertical_Line sr) is closed & ( L~ US) is_an_arc_of (Wmin,Emax) by A15, A10, JORDAN6: 30, TOPREAL1: 25;

              ( First_Point (( L~ US),(US /. 1),(US /. ( len US)),( Vertical_Line sr))) in ( rng US) by A1, A15, A10, Th47;

              then

               A40: (( L~ ( mid (US,1,(FiP .. US)))) /\ ( Vertical_Line sr)) = {FiP} by A15, A10, A39, A38, A37, Th50, JORDAN6: 49;

              

               A41: ( mid (US,1,(FiP .. US))) = (US | (FiP .. US)) & (US | ( Seg (FiP .. US))) = (US | (FiP .. US)) by A13, FINSEQ_1:def 15, FINSEQ_6: 116;

              assume ((US /. (k + 1)) `1 ) = sr;

              then (US /. (k + 1)) in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = sr };

              then

               A42: (US /. (k + 1)) in ( Vertical_Line sr) by JORDAN6:def 6;

              

               A43: (k + 1) in ( dom US) by A16, A19, FINSEQ_3: 25;

              (k + 1) in ( Seg (FiP .. US)) by A16, A17, FINSEQ_1: 1;

              then (US /. (k + 1)) in ( rng ( mid (US,1,(FiP .. US)))) by A41, A43, PARTFUN2: 18;

              then (US /. (k + 1)) in {FiP} by A42, A35, A40, XBOOLE_0:def 4;

              then (US /. (k + 1)) = FiP by TARSKI:def 1;

              hence contradiction by A17, A12, A43, A36, PARTFUN2: 10;

            end;

            i1 < ( Center ( Gauge (C,n))) by A6, A17, A21, A30, A26, A14, A7, A8, JORDAN1A: 18, NAT_1: 13, NAT_1: 14;

            then (i1 + 1) <= ( Center ( Gauge (C,n))) by NAT_1: 13;

            then ((US /. (k + 1)) `1 ) <= sr by A23, A26, A7, A8, A31, A32, JORDAN1A: 18;

            hence thesis by A33, XXREAL_0: 1;

          end;

            suppose i1 = (i2 + 1) & j1 = j2;

            then i2 < i1 by NAT_1: 13;

            then ((US /. (k + 1)) `1 ) <= ((US /. k) `1 ) by A21, A23, A30, A26, A28, A27, JORDAN1A: 18;

            hence thesis by A6, A17, NAT_1: 13, NAT_1: 14, XXREAL_0: 2;

          end;

            suppose i1 = i2 & j1 = (j2 + 1);

            

            then ((US /. k) `1 ) = ((( Gauge (C,n)) * (i2,1)) `1 ) by A21, A25, A30, A26, GOBOARD5: 2

            .= ((US /. (k + 1)) `1 ) by A23, A29, A28, A27, GOBOARD5: 2;

            hence thesis by A6, A17, NAT_1: 13, NAT_1: 14;

          end;

        end;

        hence thesis;

      end;

      

       A44: P[1]

      proof

        assume that 1 <= 1 and 1 < (FiP .. US);

        (US /. 1) = Wmin by JORDAN1F: 5;

        hence thesis by A3, EUCLID: 52;

      end;

      

       A45: for k be non zero Nat holds P[k] from NAT_1:sch 10( A44, A5);

      let k be Nat;

      assume 1 <= k & k < (( First_Point (( L~ ( Upper_Seq (C,n))),( W-min ( L~ ( Cage (C,n)))),( E-max ( L~ ( Cage (C,n)))),( Vertical_Line ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)))) .. ( Upper_Seq (C,n)));

      hence thesis by A45;

    end;

    theorem :: JORDAN1G:52

    

     Th52: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for n be Nat st n > 0 holds for k be Nat st 1 <= k & k < (( First_Point (( L~ ( Rev ( Lower_Seq (C,n)))),( W-min ( L~ ( Cage (C,n)))),( E-max ( L~ ( Cage (C,n)))),( Vertical_Line ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)))) .. ( Rev ( Lower_Seq (C,n)))) holds ((( Rev ( Lower_Seq (C,n))) /. k) `1 ) < ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let n be Nat;

      assume

       A1: n > 0 ;

      set LS = ( Lower_Seq (C,n));

      set sr = ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2);

      set Ebo = ( E-bound ( L~ ( Cage (C,n))));

      set Wbo = ( W-bound ( L~ ( Cage (C,n))));

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      set RLS = ( Rev LS);

      set FiP = ( First_Point (( L~ RLS),Wmin,Emax,( Vertical_Line sr)));

      set LaP = ( Last_Point (( L~ LS),Emax,Wmin,( Vertical_Line sr)));

      

       A2: ( L~ RLS) = ( L~ LS) by SPPOL_2: 22;

      

       A3: ( len RLS) = ( len LS) by FINSEQ_5:def 3;

      defpred P[ Nat] means 1 <= $1 & $1 < (FiP .. RLS) implies ((RLS /. $1) `1 ) < sr;

      

       A4: ( rng RLS) = ( rng LS) by FINSEQ_5: 57;

      

       A5: Wbo < Ebo by SPRECT_1: 31;

      then

       A6: Wbo < sr by XREAL_1: 226;

      

       A7: sr < Ebo by A5, XREAL_1: 226;

      

       A8: for k be non zero Nat st P[k] holds P[(k + 1)]

      proof

        

         A9: Wbo <= Ebo by SPRECT_1: 21;

        then Wbo <= sr by JORDAN6: 1;

        then

         A10: (Wmin `1 ) <= sr by EUCLID: 52;

        sr <= Ebo by A9, JORDAN6: 1;

        then

         A11: sr <= (Emax `1 ) by EUCLID: 52;

        

         A12: (RLS /. ( len RLS)) = (LS /. 1) by A3, FINSEQ_5: 65

        .= Emax by JORDAN1F: 6;

        set GC1 = (( Gauge (C,n)) * (( Center ( Gauge (C,n))),1));

        let k be non zero Nat;

        assume

         A13: 1 <= k & k < (FiP .. RLS) implies ((RLS /. k) `1 ) < sr;

        4 <= ( len ( Gauge (C,n))) by JORDAN8: 10;

        then 1 <= ( len ( Gauge (C,n))) by XXREAL_0: 2;

        then

         A14: 1 <= ( width ( Gauge (C,n))) by JORDAN8:def 1;

        

        then

         A15: (GC1 `1 ) = ((( W-bound C) + ( E-bound C)) / 2) by A1, Th35

        .= sr by Th33;

        

         A16: (LS /. 1) = Emax & (LS /. ( len LS)) = Wmin by JORDAN1F: 6, JORDAN1F: 8;

        then

         A17: ( L~ LS) is_an_arc_of (Emax,Wmin) by TOPREAL1: 25;

        

         A18: 1 <= ( Center ( Gauge (C,n))) by JORDAN1B: 11;

        

         A19: (RLS /. 1) = (LS /. ( len LS)) by FINSEQ_5: 65

        .= Wmin by JORDAN1F: 8;

        ( L~ LS) is_an_arc_of (Wmin,Emax) by A16, JORDAN5B: 14, TOPREAL1: 25;

        then ( L~ LS) meets ( Vertical_Line sr) & (( L~ LS) /\ ( Vertical_Line sr)) is closed by A10, A11, JORDAN6: 49;

        then

         A20: FiP = LaP by A2, A17, JORDAN5C: 18;

        then

         A21: (FiP .. RLS) in ( dom RLS) by A1, A4, Th48, FINSEQ_4: 20;

        then

         A22: 1 <= (FiP .. RLS) by FINSEQ_3: 25;

        

         A23: k >= 1 by NAT_1: 14;

        reconsider kk = k as Nat;

        assume that

         A24: 1 <= (k + 1) and

         A25: (k + 1) < (FiP .. RLS);

        

         A26: (FiP .. RLS) <= ( len RLS) by A21, FINSEQ_3: 25;

        then

         A27: (k + 1) <= ( len RLS) by A25, XXREAL_0: 2;

        LS is_sequence_on ( Gauge (C,n)) by Th5;

        then RLS is_sequence_on ( Gauge (C,n)) by JORDAN9: 5;

        then

        consider i1,j1,i2,j2 be Nat such that

         A28: [i1, j1] in ( Indices ( Gauge (C,n))) and

         A29: (RLS /. kk) = (( Gauge (C,n)) * (i1,j1)) and

         A30: [i2, j2] in ( Indices ( Gauge (C,n))) and

         A31: (RLS /. (kk + 1)) = (( Gauge (C,n)) * (i2,j2)) and

         A32: i1 = i2 & (j1 + 1) = j2 or (i1 + 1) = i2 & j1 = j2 or i1 = (i2 + 1) & j1 = j2 or i1 = i2 & j1 = (j2 + 1) by A23, A27, JORDAN8: 3;

        

         A33: 1 <= i1 by A28, MATRIX_0: 32;

        

         A34: 1 <= j1 & j1 <= ( width ( Gauge (C,n))) by A28, MATRIX_0: 32;

        

         A35: i2 <= ( len ( Gauge (C,n))) by A30, MATRIX_0: 32;

        

         A36: i1 <= ( len ( Gauge (C,n))) by A28, MATRIX_0: 32;

        

         A37: j2 <= ( width ( Gauge (C,n))) by A30, MATRIX_0: 32;

        

         A38: 1 <= i2 & 1 <= j2 by A30, MATRIX_0: 32;

        

         A39: ( Center ( Gauge (C,n))) <= ( len ( Gauge (C,n))) & (i1 + 1) >= 1 by JORDAN1B: 13, NAT_1: 11;

        now

          per cases by A32;

            suppose i1 = i2 & (j1 + 1) = j2;

            

            then ((RLS /. k) `1 ) = ((( Gauge (C,n)) * (i2,1)) `1 ) by A29, A33, A36, A34, GOBOARD5: 2

            .= ((RLS /. (k + 1)) `1 ) by A31, A35, A38, A37, GOBOARD5: 2;

            hence thesis by A13, A25, NAT_1: 13, NAT_1: 14;

          end;

            suppose

             A40: (i1 + 1) = i2 & j1 = j2;

             A41:

            now

               A42:

              now

                assume (RLS /. 1) in ( Vertical_Line sr);

                then (Wmin `1 ) = sr by A19, JORDAN6: 31;

                hence contradiction by A6, EUCLID: 52;

              end;

              assume ((RLS /. (k + 1)) `1 ) = sr;

              then (RLS /. (k + 1)) in { p where p be Point of ( TOP-REAL 2) : (p `1 ) = sr };

              then

               A43: (RLS /. (k + 1)) in ( Vertical_Line sr) by JORDAN6:def 6;

              

               A44: sr <= (Emax `1 ) by A7, EUCLID: 52;

              ( L~ RLS) is_an_arc_of (Wmin,Emax) & (Wmin `1 ) <= sr by A6, A19, A12, EUCLID: 52, TOPREAL1: 25;

              then

               A45: ( L~ RLS) meets ( Vertical_Line sr) by A44, JORDAN6: 49;

              

               A46: (RLS /. (FiP .. RLS)) = FiP by A1, A4, A20, Th48, FINSEQ_5: 38;

              

               A47: (k + 1) >= (1 + 1) by A23, XREAL_1: 7;

              ( len ( mid (RLS,1,(FiP .. RLS)))) = (((FiP .. RLS) -' 1) + 1) by A22, A26, JORDAN4: 8

              .= (FiP .. RLS) by A22, XREAL_1: 235;

              then

               A48: ( rng ( mid (RLS,1,(FiP .. RLS)))) c= ( L~ ( mid (RLS,1,(FiP .. RLS)))) by A25, A47, SPPOL_2: 18, XXREAL_0: 2;

              

               A49: (k + 1) in ( dom RLS) by A24, A27, FINSEQ_3: 25;

              ( Vertical_Line sr) is closed & RLS is being_S-Seq by JORDAN6: 30;

              then

               A50: (( L~ ( mid (RLS,1,(FiP .. RLS)))) /\ ( Vertical_Line sr)) = {FiP} by A1, A4, A20, A19, A12, A45, A42, Th48, Th50;

              

               A51: ( mid (RLS,1,(FiP .. RLS))) = (RLS | (FiP .. RLS)) & (RLS | ( Seg (FiP .. RLS))) = (RLS | (FiP .. RLS)) by A22, FINSEQ_1:def 15, FINSEQ_6: 116;

              (k + 1) in ( Seg (FiP .. RLS)) by A24, A25, FINSEQ_1: 1;

              then (RLS /. (k + 1)) in ( rng ( mid (RLS,1,(FiP .. RLS)))) by A51, A49, PARTFUN2: 18;

              then (RLS /. (k + 1)) in {FiP} by A43, A48, A50, XBOOLE_0:def 4;

              then (RLS /. (k + 1)) = FiP by TARSKI:def 1;

              hence contradiction by A25, A21, A49, A46, PARTFUN2: 10;

            end;

            i1 < ( Center ( Gauge (C,n))) by A13, A25, A29, A36, A34, A18, A14, A15, JORDAN1A: 18, NAT_1: 13, NAT_1: 14;

            then (i1 + 1) <= ( Center ( Gauge (C,n))) by NAT_1: 13;

            then ((RLS /. (k + 1)) `1 ) <= sr by A31, A34, A14, A15, A39, A40, JORDAN1A: 18;

            hence thesis by A41, XXREAL_0: 1;

          end;

            suppose i1 = (i2 + 1) & j1 = j2;

            then i2 < i1 by NAT_1: 13;

            then ((RLS /. (k + 1)) `1 ) <= ((RLS /. k) `1 ) by A29, A31, A36, A34, A38, A37, JORDAN1A: 18;

            hence thesis by A13, A25, NAT_1: 13, NAT_1: 14, XXREAL_0: 2;

          end;

            suppose i1 = i2 & j1 = (j2 + 1);

            

            then ((RLS /. k) `1 ) = ((( Gauge (C,n)) * (i2,1)) `1 ) by A29, A33, A36, A34, GOBOARD5: 2

            .= ((RLS /. (k + 1)) `1 ) by A31, A35, A38, A37, GOBOARD5: 2;

            hence thesis by A13, A25, NAT_1: 13, NAT_1: 14;

          end;

        end;

        hence thesis;

      end;

      

       A52: P[1]

      proof

        assume that 1 <= 1 and 1 < (FiP .. RLS);

        (RLS /. 1) = (LS /. ( len LS)) by FINSEQ_5: 65

        .= Wmin by JORDAN1F: 8;

        hence thesis by A6, EUCLID: 52;

      end;

      

       A53: for k be non zero Nat holds P[k] from NAT_1:sch 10( A52, A8);

      let k be Nat;

      assume 1 <= k & k < (( First_Point (( L~ ( Rev ( Lower_Seq (C,n)))),( W-min ( L~ ( Cage (C,n)))),( E-max ( L~ ( Cage (C,n)))),( Vertical_Line ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)))) .. ( Rev ( Lower_Seq (C,n))));

      hence thesis by A53;

    end;

    theorem :: JORDAN1G:53

    

     Th53: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for n be Nat st n > 0 holds for q be Point of ( TOP-REAL 2) holds q in ( rng ( mid (( Upper_Seq (C,n)),2,(( First_Point (( L~ ( Upper_Seq (C,n))),( W-min ( L~ ( Cage (C,n)))),( E-max ( L~ ( Cage (C,n)))),( Vertical_Line ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)))) .. ( Upper_Seq (C,n)))))) implies (q `1 ) <= ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let n be Nat;

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      set Wbo = ( W-bound ( L~ ( Cage (C,n))));

      set Ebo = ( E-bound ( L~ ( Cage (C,n))));

      set sr = ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2);

      set US = ( Upper_Seq (C,n));

      set FiP = ( First_Point (( L~ US),Wmin,Emax,( Vertical_Line sr)));

      

       A1: (US /. 1) = Wmin by JORDAN1F: 5;

      (US /. ( len US)) = Emax by JORDAN1F: 7;

      then

       A2: ( L~ US) is_an_arc_of (Wmin,Emax) by A1, TOPREAL1: 25;

      assume

       A3: n > 0 ;

      then

       A4: FiP in ( rng US) by Th47;

      then

       A5: (FiP .. US) in ( dom US) by FINSEQ_4: 20;

      then

       A6: (FiP .. US) <= ( len US) by FINSEQ_3: 25;

      

       A7: Wbo < Ebo by SPRECT_1: 31;

      then

       A8: Wbo < sr by XREAL_1: 226;

      sr < Ebo by A7, XREAL_1: 226;

      then

       A9: sr <= (Emax `1 ) by EUCLID: 52;

      (Wmin `1 ) <= sr by A8, EUCLID: 52;

      then ( L~ US) meets ( Vertical_Line sr) & (( L~ US) /\ ( Vertical_Line sr)) is closed by A2, A9, JORDAN6: 49;

      then FiP in (( L~ US) /\ ( Vertical_Line sr)) by A2, JORDAN5C:def 1;

      then FiP in ( Vertical_Line sr) by XBOOLE_0:def 4;

      then

       A10: (FiP `1 ) = sr by JORDAN6: 31;

      

       A11: Wmin in ( rng US) by A1, FINSEQ_6: 42;

       A12:

      now

        assume (FiP .. US) = 1;

        

        then (FiP .. US) = ((US /. 1) .. US) by FINSEQ_6: 43

        .= (Wmin .. US) by JORDAN1F: 5;

        then FiP = Wmin by A4, A11, FINSEQ_5: 9;

        hence contradiction by A8, A10, EUCLID: 52;

      end;

      1 <= (FiP .. US) by A5, FINSEQ_3: 25;

      then (FiP .. US) > 1 by A12, XXREAL_0: 1;

      then

       A13: ((1 + 1) + 0 ) <= (FiP .. US) by NAT_1: 13;

      then ((FiP .. US) - 2) >= 0 by XREAL_1: 19;

      then ((FiP .. US) -' 2) = ((FiP .. US) - 2) by XREAL_0:def 2;

      then

       A14: ( len ( mid (US,2,(FiP .. US)))) = (((FiP .. US) - 2) + 1) by A6, A13, JORDAN4: 8;

      let q be Point of ( TOP-REAL 2);

      assume q in ( rng ( mid (US,2,(FiP .. US))));

      then

      consider k be Element of NAT such that

       A15: k in ( dom ( mid (US,2,(FiP .. US)))) and

       A16: q = (( mid (US,2,(FiP .. US))) /. k) by PARTFUN2: 2;

      (k + 2) >= (1 + 1) by NAT_1: 11;

      then

       A17: ((k + 2) - 1) >= ((1 + 1) - 1) by XREAL_1: 9;

      ( len US) >= 3 by JORDAN1E: 15;

      then ( len US) >= 2 by XXREAL_0: 2;

      then 2 in ( dom US) by FINSEQ_3: 25;

      

      then

       A18: (( mid (US,2,(FiP .. US))) /. k) = (US /. ((k + 2) -' 1)) by A15, A5, A13, SPRECT_2: 3

      .= (US /. (k + (2 - 1))) by A17, XREAL_0:def 2;

      k <= ( len ( mid (US,2,(FiP .. US)))) by A15, FINSEQ_3: 25;

      then k < ((((FiP .. US) - 2) + 1) + 1) by A14, NAT_1: 13;

      then

       A19: (k + 1) <= (FiP .. US) by NAT_1: 13;

      per cases by A19, XXREAL_0: 1;

        suppose (k + 1) < (FiP .. US);

        hence thesis by A3, A16, A18, Th51, NAT_1: 11;

      end;

        suppose (k + 1) = (FiP .. US);

        hence thesis by A16, A4, A10, A18, FINSEQ_5: 38;

      end;

    end;

    theorem :: JORDAN1G:54

    

     Th54: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for n be Nat st n > 0 holds (( First_Point (( L~ ( Upper_Seq (C,n))),( W-min ( L~ ( Cage (C,n)))),( E-max ( L~ ( Cage (C,n)))),( Vertical_Line ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)))) `2 ) > (( Last_Point (( L~ ( Lower_Seq (C,n))),( E-max ( L~ ( Cage (C,n)))),( W-min ( L~ ( Cage (C,n)))),( Vertical_Line ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)))) `2 )

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let n be Nat;

      set sr = ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2);

      set Wmin = ( W-min ( L~ ( Cage (C,n))));

      set Emax = ( E-max ( L~ ( Cage (C,n))));

      set Nbo = ( N-bound ( L~ ( Cage (C,n))));

      set Ebo = ( E-bound ( L~ ( Cage (C,n))));

      set Sbo = ( S-bound ( L~ ( Cage (C,n))));

      set Wbo = ( W-bound ( L~ ( Cage (C,n))));

      set SW = ( SW-corner ( L~ ( Cage (C,n))));

      set FiP = ( First_Point (( L~ ( Upper_Seq (C,n))),Wmin,Emax,( Vertical_Line sr)));

      set LaP = ( Last_Point (( L~ ( Lower_Seq (C,n))),Emax,Wmin,( Vertical_Line sr)));

      set g = (( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n))))) ^ <* |[Ebo, (FiP `2 )]|*>);

      set h = (( <*SW*> ^ (( Rev ( Lower_Seq (C,n))) -: LaP)) ^ <* |[sr, Nbo]|*>);

      

       A1: (( Upper_Seq (C,n)) /. 1) = ( W-min ( L~ ( Cage (C,n)))) by JORDAN1F: 5;

      

       A2: Wbo <= Ebo by SPRECT_1: 21;

      then Wbo <= sr by JORDAN6: 1;

      then

       A3: (Wmin `1 ) <= sr by EUCLID: 52;

      sr <= Ebo by A2, JORDAN6: 1;

      then

       A4: sr <= (Emax `1 ) by EUCLID: 52;

      set GCw = (( Gauge (C,n)) * (( Center ( Gauge (C,n))),( width ( Gauge (C,n)))));

      

       A5: 1 <= ( Center ( Gauge (C,n))) by JORDAN1B: 11;

      ( len ( Gauge (C,n))) = ( width ( Gauge (C,n))) by JORDAN8:def 1;

      then

       A6: (GCw `2 ) = Nbo by A5, JORDAN1A: 70, JORDAN1B: 13;

      

       A7: (SW `2 ) <= (Wmin `2 ) by PSCOMP_1: 30;

      

       A8: ( |[sr, Nbo]| `2 ) = Nbo by EUCLID: 52;

      set RevL = (( Rev ( Lower_Seq (C,n))) -: LaP);

      

       A9: <* |[Ebo, (FiP `2 )]|*> is one-to-one & <* |[Ebo, (FiP `2 )]|*> is special by FINSEQ_3: 93;

      

       A10: ( rng (( Rev ( Lower_Seq (C,n))) -: LaP)) c= ( rng ( Rev ( Lower_Seq (C,n)))) by FINSEQ_5: 48;

      

       A11: (( Lower_Seq (C,n)) /. 1) = ( E-max ( L~ ( Cage (C,n)))) & (( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) = ( W-min ( L~ ( Cage (C,n)))) by JORDAN1F: 6, JORDAN1F: 8;

      then

       A12: ( L~ ( Lower_Seq (C,n))) is_an_arc_of (Emax,Wmin) by TOPREAL1: 25;

      

       A13: 4 <= ( len ( Gauge (C,n))) by JORDAN8: 10;

      then

       A14: ( len ( Gauge (C,n))) >= 3 by XXREAL_0: 2;

      

       A15: Wbo < Ebo by SPRECT_1: 31;

      then

       A16: Wbo < sr by XREAL_1: 226;

      ( L~ ( Lower_Seq (C,n))) is_an_arc_of (Wmin,Emax) by A11, JORDAN5B: 14, TOPREAL1: 25;

      then

       A17: ( L~ ( Lower_Seq (C,n))) meets ( Vertical_Line sr) & (( L~ ( Lower_Seq (C,n))) /\ ( Vertical_Line sr)) is closed by A3, A4, JORDAN6: 49;

      then

       A18: LaP in (( L~ ( Lower_Seq (C,n))) /\ ( Vertical_Line sr)) by A12, JORDAN5C:def 2;

      then

       A19: LaP in ( L~ ( Lower_Seq (C,n))) by XBOOLE_0:def 4;

      then LaP in (( L~ ( Upper_Seq (C,n))) \/ ( L~ ( Lower_Seq (C,n)))) by XBOOLE_0:def 3;

      then

       A20: LaP in ( L~ ( Cage (C,n))) by JORDAN1E: 13;

      assume

       A21: n > 0 ;

      then

       A22: FiP in ( rng ( Upper_Seq (C,n))) by Th47;

      then

       A23: (FiP .. ( Upper_Seq (C,n))) in ( dom ( Upper_Seq (C,n))) by FINSEQ_4: 20;

      then

       A24: 1 <= (FiP .. ( Upper_Seq (C,n))) by FINSEQ_3: 25;

      1 <= ( len ( Gauge (C,n))) by A13, XXREAL_0: 2;

      then 1 <= ( width ( Gauge (C,n))) by JORDAN8:def 1;

      

      then (GCw `1 ) = ((( W-bound C) + ( E-bound C)) / 2) by A21, Th35

      .= sr by Th33;

      then GCw = |[sr, Nbo]| by A6, EUCLID: 53;

      then not |[sr, Nbo]| in ( rng ( Lower_Seq (C,n))) by A5, A14, Th43, JORDAN1B: 15;

      then not |[sr, Nbo]| in ( rng ( Rev ( Lower_Seq (C,n)))) by FINSEQ_5: 57;

      then

       A25: not |[sr, Nbo]| in ( rng (( Rev ( Lower_Seq (C,n))) -: LaP)) by A10;

      (SW `2 ) = Sbo by EUCLID: 52;

      then |[sr, Nbo]| <> SW by A8, SPRECT_1: 32;

      then not |[sr, Nbo]| in {SW} by TARSKI:def 1;

      then not |[sr, Nbo]| in ( rng <*SW*>) by FINSEQ_1: 38;

      then not |[sr, Nbo]| in (( rng <*SW*>) \/ ( rng (( Rev ( Lower_Seq (C,n))) -: LaP))) by A25, XBOOLE_0:def 3;

      then not |[sr, Nbo]| in ( rng ( <*SW*> ^ (( Rev ( Lower_Seq (C,n))) -: LaP))) by FINSEQ_1: 31;

      then ( rng ( <*SW*> ^ (( Rev ( Lower_Seq (C,n))) -: LaP))) misses { |[sr, Nbo]|} by ZFMISC_1: 50;

      then (( rng ( <*SW*> ^ (( Rev ( Lower_Seq (C,n))) -: LaP))) /\ { |[sr, Nbo]|}) = {} ;

      then (( rng ( <*SW*> ^ (( Rev ( Lower_Seq (C,n))) -: LaP))) /\ ( rng <* |[sr, Nbo]|*>)) = {} by FINSEQ_1: 38;

      then

       A26: ( rng ( <*SW*> ^ (( Rev ( Lower_Seq (C,n))) -: LaP))) misses ( rng <* |[sr, Nbo]|*>);

      LaP in ( rng ( Lower_Seq (C,n))) by A21, Th48;

      then

       A27: LaP in ( rng ( Rev ( Lower_Seq (C,n)))) by FINSEQ_5: 57;

      then

       A28: RevL is non empty by FINSEQ_5: 47;

      

       A29: ( len RevL) = (LaP .. ( Rev ( Lower_Seq (C,n)))) by A27, FINSEQ_5: 42;

      

       A30: (( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) = ( E-max ( L~ ( Cage (C,n)))) by JORDAN1F: 7;

      then

       A31: ( L~ ( Upper_Seq (C,n))) is_an_arc_of (Wmin,Emax) by A1, TOPREAL1: 25;

      

       A32: sr < Ebo by A15, XREAL_1: 226;

      then

       A33: sr <= (Emax `1 ) by EUCLID: 52;

      (Wmin `1 ) <= sr by A16, EUCLID: 52;

      then ( L~ ( Upper_Seq (C,n))) meets ( Vertical_Line sr) & (( L~ ( Upper_Seq (C,n))) /\ ( Vertical_Line sr)) is closed by A31, A33, JORDAN6: 49;

      then

       A34: FiP in (( L~ ( Upper_Seq (C,n))) /\ ( Vertical_Line sr)) by A31, JORDAN5C:def 1;

      then

       A35: FiP in ( L~ ( Upper_Seq (C,n))) by XBOOLE_0:def 4;

      then FiP in (( L~ ( Upper_Seq (C,n))) \/ ( L~ ( Lower_Seq (C,n)))) by XBOOLE_0:def 3;

      then

       A36: FiP in ( L~ ( Cage (C,n))) by JORDAN1E: 13;

      now

        let m be Nat;

        assume m in ( dom <* |[Ebo, (FiP `2 )]|*>);

        then m in ( Seg 1) by FINSEQ_1: 38;

        then m = 1 by FINSEQ_1: 2, TARSKI:def 1;

        then

         A37: ( <* |[Ebo, (FiP `2 )]|*> /. m) = |[Ebo, (FiP `2 )]| by FINSEQ_4: 16;

        then (( <* |[Ebo, (FiP `2 )]|*> /. m) `1 ) = Ebo by EUCLID: 52;

        hence ( W-bound ( L~ ( Cage (C,n)))) <= (( <* |[Ebo, (FiP `2 )]|*> /. m) `1 ) & (( <* |[Ebo, (FiP `2 )]|*> /. m) `1 ) <= ( E-bound ( L~ ( Cage (C,n)))) by SPRECT_1: 21;

        (( <* |[Ebo, (FiP `2 )]|*> /. m) `2 ) = (FiP `2 ) by A37, EUCLID: 52;

        hence ( S-bound ( L~ ( Cage (C,n)))) <= (( <* |[Ebo, (FiP `2 )]|*> /. m) `2 ) & (( <* |[Ebo, (FiP `2 )]|*> /. m) `2 ) <= ( N-bound ( L~ ( Cage (C,n)))) by A36, PSCOMP_1: 24;

      end;

      then

       A38: <* |[Ebo, (FiP `2 )]|*> is_in_the_area_of ( Cage (C,n)) by SPRECT_2:def 1;

      

       A39: FiP in ( Vertical_Line sr) by A34, XBOOLE_0:def 4;

      then

       A40: (FiP `1 ) = sr by JORDAN6: 31;

      now

        assume (( rng ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n)))))) /\ { |[Ebo, (FiP `2 )]|}) <> {} ;

        then

        consider x be object such that

         A41: x in (( rng ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n)))))) /\ { |[Ebo, (FiP `2 )]|}) by XBOOLE_0:def 1;

        x in ( rng ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n)))))) & x in { |[Ebo, (FiP `2 )]|} by A41, XBOOLE_0:def 4;

        then |[Ebo, (FiP `2 )]| in ( rng ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n)))))) by TARSKI:def 1;

        then ( |[Ebo, (FiP `2 )]| `1 ) <= sr by A21, Th53;

        hence contradiction by A32, EUCLID: 52;

      end;

      then ( rng ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n)))))) misses { |[Ebo, (FiP `2 )]|};

      then

       A42: ( rng ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n)))))) misses ( rng <* |[Ebo, (FiP `2 )]|*>) by FINSEQ_1: 38;

      

       A43: (FiP .. ( Upper_Seq (C,n))) <= ( len ( Upper_Seq (C,n))) by A23, FINSEQ_3: 25;

      LaP in ( Vertical_Line sr) by A18, XBOOLE_0:def 4;

      then

       A44: (LaP `1 ) = sr by JORDAN6: 31;

       A45:

      now

        assume (FiP `2 ) = (LaP `2 );

        then FiP = LaP by A40, A44, TOPREAL3: 6;

        then FiP in (( L~ ( Upper_Seq (C,n))) /\ ( L~ ( Lower_Seq (C,n)))) by A35, A19, XBOOLE_0:def 4;

        then FiP in {Wmin, Emax} by JORDAN1E: 16;

        then FiP = Wmin or FiP = Emax by TARSKI:def 2;

        hence contradiction by A16, A32, A40, EUCLID: 52;

      end;

      ( len ( Upper_Seq (C,n))) >= 3 by JORDAN1E: 15;

      then

       A46: ( len ( Upper_Seq (C,n))) > 2 by XXREAL_0: 2;

      then

       A47: 2 in ( dom ( Upper_Seq (C,n))) by FINSEQ_3: 25;

      

      then

       A48: ((( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n))))) /. ( len ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n))))))) `2 ) = ((( Upper_Seq (C,n)) /. (FiP .. ( Upper_Seq (C,n)))) `2 ) by A23, SPRECT_2: 9

      .= (FiP `2 ) by A22, FINSEQ_5: 38

      .= ( |[Ebo, (FiP `2 )]| `2 ) by EUCLID: 52

      .= (( <* |[Ebo, (FiP `2 )]|*> /. 1) `2 ) by FINSEQ_4: 16;

      2 <> (FiP .. ( Upper_Seq (C,n)))

      proof

        assume 2 = (FiP .. ( Upper_Seq (C,n)));

        then (( Upper_Seq (C,n)) /. 2) = FiP by A22, FINSEQ_5: 38;

        then (FiP `1 ) = Wbo by Th31;

        then Wbo = sr by A39, JORDAN6: 31;

        hence contradiction by SPRECT_1: 31;

      end;

      then ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n))))) is being_S-Seq by A46, A24, A43, JORDAN3: 6;

      then

      reconsider g as one-to-one special FinSequence of ( TOP-REAL 2) by A42, A48, A9, FINSEQ_3: 91, GOBOARD2: 8;

      ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n))))) is_in_the_area_of ( Cage (C,n)) by A47, A23, JORDAN1E: 17, SPRECT_2: 22;

      then

       A49: g is_in_the_area_of ( Cage (C,n)) by A38, SPRECT_2: 24;

      

       A50: ((g /. ( len g)) `1 ) = (( <* |[Ebo, (FiP `2 )]|*> /. ( len <* |[Ebo, (FiP `2 )]|*>)) `1 ) by SPRECT_3: 1

      .= (( <* |[Ebo, (FiP `2 )]|*> /. 1) `1 ) by FINSEQ_1: 39

      .= ( |[Ebo, (FiP `2 )]| `1 ) by FINSEQ_4: 16

      .= ( E-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

      

       A51: 1 <= ( len ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n)))))) by A47, A23, SPRECT_2: 5;

      then 1 in ( dom ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n)))))) by FINSEQ_3: 25;

      

      then ((g /. 1) `1 ) = ((( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n))))) /. 1) `1 ) by FINSEQ_4: 68

      .= ((( Upper_Seq (C,n)) /. 2) `1 ) by A47, A23, SPRECT_2: 8

      .= ( W-bound ( L~ ( Cage (C,n)))) by Th31;

      then

       A52: g is_a_h.c._for ( Cage (C,n)) by A49, A50, SPRECT_2:def 2;

      assume (FiP `2 ) <= (LaP `2 );

      then

       A53: (FiP `2 ) < (LaP `2 ) by A45, XXREAL_0: 1;

      

       A54: ( rng ( Lower_Seq (C,n))) c= ( rng ( Cage (C,n))) by Th39;

      now

        per cases ;

          suppose

           A55: SW <> Wmin;

           not SW in ( rng ( Lower_Seq (C,n)))

          proof

            (SW `1 ) = (Wmin `1 ) by PSCOMP_1: 29;

            then

             A56: (SW `2 ) <> (Wmin `2 ) by A55, TOPREAL3: 6;

            assume SW in ( rng ( Lower_Seq (C,n)));

            then

             A57: SW in ( rng ( Cage (C,n))) by A54;

            ( len ( Cage (C,n))) > 4 by GOBOARD7: 34;

            then

             A58: ( rng ( Cage (C,n))) c= ( L~ ( Cage (C,n))) by SPPOL_2: 18, XXREAL_0: 2;

            (SW `1 ) = ( W-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

            then SW in ( W-most ( L~ ( Cage (C,n)))) by A57, A58, SPRECT_2: 12;

            then (Wmin `2 ) <= (SW `2 ) by PSCOMP_1: 31;

            hence contradiction by A7, A56, XXREAL_0: 1;

          end;

          then not SW in ( rng ( Rev ( Lower_Seq (C,n)))) by FINSEQ_5: 57;

          then not SW in ( rng (( Rev ( Lower_Seq (C,n))) -: LaP)) by A10;

          then {SW} misses ( rng (( Rev ( Lower_Seq (C,n))) -: LaP)) by ZFMISC_1: 50;

          then ( {SW} /\ ( rng (( Rev ( Lower_Seq (C,n))) -: LaP))) = {} ;

          then (( rng <*SW*>) /\ ( rng (( Rev ( Lower_Seq (C,n))) -: LaP))) = {} by FINSEQ_1: 38;

          then

           A59: ( rng <*SW*>) misses ( rng (( Rev ( Lower_Seq (C,n))) -: LaP));

           <*SW*> is one-to-one by FINSEQ_3: 93;

          then

           A60: ( <*SW*> ^ (( Rev ( Lower_Seq (C,n))) -: LaP)) is one-to-one by A59, FINSEQ_3: 91;

          set FiP2 = ( First_Point (( L~ ( Lower_Seq (C,n))),Wmin,Emax,( Vertical_Line sr)));

          set midU = ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n)))));

          reconsider RevLS = ( Rev ( Lower_Seq (C,n))) as special FinSequence of ( TOP-REAL 2);

          (( <*SW*> /. ( len <*SW*>)) `1 ) = (( <*SW*> /. 1) `1 ) by FINSEQ_1: 39

          .= (SW `1 ) by FINSEQ_4: 16

          .= ( W-bound ( L~ ( Cage (C,n)))) by EUCLID: 52

          .= (( W-min ( L~ ( Cage (C,n)))) `1 ) by EUCLID: 52

          .= ((( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) `1 ) by JORDAN1F: 8

          .= ((( Rev ( Lower_Seq (C,n))) /. 1) `1 ) by FINSEQ_5: 65

          .= (((( Rev ( Lower_Seq (C,n))) -: LaP) /. 1) `1 ) by A27, FINSEQ_5: 44;

          then

           A61: ( <*SW*> ^ (RevLS -: LaP)) is special by GOBOARD2: 8;

          (( Rev ( Lower_Seq (C,n))) -: LaP) is non empty by A27, FINSEQ_5: 47;

          

          then

           A62: ((( <*SW*> ^ (( Rev ( Lower_Seq (C,n))) -: LaP)) /. ( len ( <*SW*> ^ (( Rev ( Lower_Seq (C,n))) -: LaP)))) `1 ) = (((( Rev ( Lower_Seq (C,n))) -: LaP) /. ( len (( Rev ( Lower_Seq (C,n))) -: LaP))) `1 ) by SPRECT_3: 1

          .= (((( Rev ( Lower_Seq (C,n))) -: LaP) /. (LaP .. ( Rev ( Lower_Seq (C,n))))) `1 ) by A27, FINSEQ_5: 42

          .= (LaP `1 ) by A27, FINSEQ_5: 45

          .= ( |[sr, Nbo]| `1 ) by A44, EUCLID: 52

          .= (( <* |[sr, Nbo]|*> /. 1) `1 ) by FINSEQ_4: 16;

           <* |[sr, Nbo]|*> is one-to-one & <* |[sr, Nbo]|*> is special by FINSEQ_3: 93;

          then

          reconsider h as one-to-one special FinSequence of ( TOP-REAL 2) by A26, A60, A62, A61, FINSEQ_3: 91, GOBOARD2: 8;

          

           A63: ( |[Ebo, (FiP `2 )]| `1 ) = Ebo by EUCLID: 52;

          now

            let m be Nat;

            assume m in ( dom <*SW*>);

            then m in ( Seg 1) by FINSEQ_1: 38;

            then m = 1 by FINSEQ_1: 2, TARSKI:def 1;

            then

             A64: ( <*SW*> /. m) = SW by FINSEQ_4: 16;

            then (( <*SW*> /. m) `1 ) = Wbo by EUCLID: 52;

            hence ( W-bound ( L~ ( Cage (C,n)))) <= (( <*SW*> /. m) `1 ) & (( <*SW*> /. m) `1 ) <= ( E-bound ( L~ ( Cage (C,n)))) by SPRECT_1: 21;

            (( <*SW*> /. m) `2 ) = Sbo by A64, EUCLID: 52;

            hence ( S-bound ( L~ ( Cage (C,n)))) <= (( <*SW*> /. m) `2 ) & (( <*SW*> /. m) `2 ) <= ( N-bound ( L~ ( Cage (C,n)))) by SPRECT_1: 22;

          end;

          then

           A65: <*SW*> is_in_the_area_of ( Cage (C,n)) by SPRECT_2:def 1;

          

           A66: (RevL /. ( len RevL)) = (RevL /. (LaP .. ( Rev ( Lower_Seq (C,n))))) by A27, FINSEQ_5: 42

          .= LaP by A27, FINSEQ_5: 45;

          now

            let m be Nat;

            

             A67: ( W-bound ( L~ ( Cage (C,n)))) <= ( E-bound ( L~ ( Cage (C,n)))) by SPRECT_1: 21;

            assume m in ( dom <* |[sr, Nbo]|*>);

            then m in ( Seg 1) by FINSEQ_1: 38;

            then m = 1 by FINSEQ_1: 2, TARSKI:def 1;

            then

             A68: ( <* |[sr, Nbo]|*> /. m) = |[sr, Nbo]| by FINSEQ_4: 16;

            then (( <* |[sr, Nbo]|*> /. m) `1 ) = sr by EUCLID: 52;

            hence ( W-bound ( L~ ( Cage (C,n)))) <= (( <* |[sr, Nbo]|*> /. m) `1 ) & (( <* |[sr, Nbo]|*> /. m) `1 ) <= ( E-bound ( L~ ( Cage (C,n)))) by A67, JORDAN6: 1;

            (( <* |[sr, Nbo]|*> /. m) `2 ) = Nbo by A68, EUCLID: 52;

            hence ( S-bound ( L~ ( Cage (C,n)))) <= (( <* |[sr, Nbo]|*> /. m) `2 ) & (( <* |[sr, Nbo]|*> /. m) `2 ) <= ( N-bound ( L~ ( Cage (C,n)))) by SPRECT_1: 22;

          end;

          then

           A69: <* |[sr, Nbo]|*> is_in_the_area_of ( Cage (C,n)) by SPRECT_2:def 1;

          

           A70: ( L~ ( Rev ( Lower_Seq (C,n)))) = ( L~ ( Lower_Seq (C,n))) & FiP2 = LaP by A12, A17, JORDAN5C: 18, SPPOL_2: 22;

          ( Rev ( Lower_Seq (C,n))) is_in_the_area_of ( Cage (C,n)) by JORDAN1E: 18, SPRECT_3: 51;

          then (( Rev ( Lower_Seq (C,n))) -: LaP) is_in_the_area_of ( Cage (C,n)) by A27, JORDAN1E: 1;

          then ( <*SW*> ^ (( Rev ( Lower_Seq (C,n))) -: LaP)) is_in_the_area_of ( Cage (C,n)) by A65, SPRECT_2: 24;

          then

           A71: h is_in_the_area_of ( Cage (C,n)) by A69, SPRECT_2: 24;

          ( len ( <*SW*> ^ (( Rev ( Lower_Seq (C,n))) -: LaP))) = (1 + ( len (( Rev ( Lower_Seq (C,n))) -: LaP))) by FINSEQ_5: 8;

          then

           A72: ( len ( <*SW*> ^ (( Rev ( Lower_Seq (C,n))) -: LaP))) >= 1 by NAT_1: 11;

          1 in ( dom h) by FINSEQ_5: 6;

          then (h /. 1) = (h . 1) by PARTFUN1:def 6;

          

          then

           A73: ((h /. 1) `2 ) = ((( <*SW*> ^ (( Rev ( Lower_Seq (C,n))) -: LaP)) /. 1) `2 ) by A72, FINSEQ_6: 109

          .= (SW `2 ) by FINSEQ_5: 15

          .= ( S-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

          

           A74: ( len h) = (( len ( <*SW*> ^ (( Rev ( Lower_Seq (C,n))) -: LaP))) + 1) by FINSEQ_2: 16;

          then

           A75: (1 + 1) <= ( len h) by A72, XREAL_1: 7;

          ( L~ ( Cage (C,n))) = (( L~ ( Upper_Seq (C,n))) \/ ( L~ ( Lower_Seq (C,n)))) by JORDAN1E: 13;

          then

           A76: ( L~ ( Upper_Seq (C,n))) c= ( L~ ( Cage (C,n))) by XBOOLE_1: 7;

          

           A77: (midU /. ( len midU)) = (( Upper_Seq (C,n)) /. (FiP .. ( Upper_Seq (C,n)))) by A47, A23, SPRECT_2: 9

          .= FiP by A22, FINSEQ_5: 38;

          

           A78: Wmin in ( rng ( Upper_Seq (C,n))) by A1, FINSEQ_6: 42;

          now

            assume (FiP .. ( Upper_Seq (C,n))) = 1;

            

            then (FiP .. ( Upper_Seq (C,n))) = ((( Upper_Seq (C,n)) /. 1) .. ( Upper_Seq (C,n))) by FINSEQ_6: 43

            .= (Wmin .. ( Upper_Seq (C,n))) by JORDAN1F: 5;

            then FiP = Wmin by A22, A78, FINSEQ_5: 9;

            hence contradiction by A16, A40, EUCLID: 52;

          end;

          then (FiP .. ( Upper_Seq (C,n))) > 1 by A24, XXREAL_0: 1;

          then

           A79: ((1 + 1) + 0 ) <= (FiP .. ( Upper_Seq (C,n))) by NAT_1: 13;

          then ((FiP .. ( Upper_Seq (C,n))) - 2) >= 0 by XREAL_1: 19;

          then ((FiP .. ( Upper_Seq (C,n))) -' 2) = ((FiP .. ( Upper_Seq (C,n))) - 2) by XREAL_0:def 2;

          

          then

           A80: ( len midU) = (((FiP .. ( Upper_Seq (C,n))) - 2) + 1) by A43, A79, JORDAN4: 8

          .= ((FiP .. ( Upper_Seq (C,n))) - (2 - 1));

          1 in ( dom RevL) by A28, FINSEQ_5: 6;

          

          then

           A81: ((RevL ^ <* |[sr, Nbo]|*>) /. 1) = (RevL /. 1) by FINSEQ_4: 68

          .= (( Rev ( Lower_Seq (C,n))) /. 1) by A27, FINSEQ_5: 44

          .= (( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) by FINSEQ_5: 65

          .= Wmin by JORDAN1F: 8;

          

           A82: (SW `2 ) <= (Wmin `2 ) by PSCOMP_1: 30;

          ( len g) = (( len ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n)))))) + 1) by FINSEQ_2: 16;

          then

           A83: (1 + 1) <= ( len g) by A51, XREAL_1: 7;

          

           A84: ( L~ g) = (( L~ midU) \/ ( LSeg ((midU /. ( len midU)), |[Ebo, (FiP `2 )]|))) by A47, A23, SPPOL_2: 19, SPRECT_2: 7;

          ( L~ ( Rev ( Lower_Seq (C,n)))) = (( L~ RevL) \/ ( L~ (( Rev ( Lower_Seq (C,n))) :- LaP))) by A27, SPPOL_2: 24;

          then ( L~ RevL) c= ( L~ ( Rev ( Lower_Seq (C,n)))) by XBOOLE_1: 7;

          then

           A85: ( L~ RevL) c= ( L~ ( Lower_Seq (C,n))) by SPPOL_2: 22;

          

           A86: (LaP `2 ) <= Nbo by A20, PSCOMP_1: 24;

          

           A87: ( |[Ebo, (FiP `2 )]| `2 ) = (FiP `2 ) by EUCLID: 52;

          then

           A88: ( LSeg (FiP, |[Ebo, (FiP `2 )]|)) is horizontal by SPPOL_1: 15;

          (LaP `1 ) = ( |[sr, Nbo]| `1 ) by A44, EUCLID: 52;

          then

           A89: ( LSeg (LaP, |[sr, Nbo]|)) is vertical by SPPOL_1: 16;

          

           A90: ( L~ midU) c= ( L~ ( Upper_Seq (C,n))) by A47, A23, SPRECT_3: 18;

          ((h /. ( len h)) `2 ) = ( |[sr, Nbo]| `2 ) by A74, FINSEQ_4: 67

          .= ( N-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

          then h is_a_v.c._for ( Cage (C,n)) by A71, A73, SPRECT_2:def 3;

          then ( L~ g) meets ( L~ h) by A52, A75, A83, SPRECT_2: 29;

          then

          consider x be object such that

           A91: x in ( L~ g) and

           A92: x in ( L~ h) by XBOOLE_0: 3;

          reconsider x as Point of ( TOP-REAL 2) by A91;

          ( L~ h) = ( L~ ( <*SW*> ^ ((( Rev ( Lower_Seq (C,n))) -: LaP) ^ <* |[sr, Nbo]|*>))) by FINSEQ_1: 32

          .= (( LSeg (SW,((RevL ^ <* |[sr, Nbo]|*>) /. 1))) \/ ( L~ (RevL ^ <* |[sr, Nbo]|*>))) by SPPOL_2: 20

          .= (( LSeg (SW,((RevL ^ <* |[sr, Nbo]|*>) /. 1))) \/ (( L~ RevL) \/ ( LSeg ((RevL /. ( len RevL)), |[sr, Nbo]|)))) by A27, FINSEQ_5: 47, SPPOL_2: 19;

          then

           A93: x in ( LSeg (SW,((RevL ^ <* |[sr, Nbo]|*>) /. 1))) or x in (( L~ RevL) \/ ( LSeg ((RevL /. ( len RevL)), |[sr, Nbo]|))) by A92, XBOOLE_0:def 3;

          

           A94: (SW `1 ) = (Wmin `1 ) by PSCOMP_1: 29;

          then

           A95: ( LSeg (SW,Wmin)) is vertical by SPPOL_1: 16;

          now

            per cases by A93, A81, A66, XBOOLE_0:def 3;

              suppose

               A96: x in ( LSeg (SW,Wmin));

              then

               A97: (x `2 ) <= (Wmin `2 ) by A82, TOPREAL1: 4;

              

               A98: (x `1 ) = (SW `1 ) by A95, A96, SPPOL_1: 41;

              then

               A99: (x `1 ) = Wbo by EUCLID: 52;

              now

                per cases by A91, A84, A77, XBOOLE_0:def 3;

                  suppose

                   A100: x in ( L~ midU);

                  then x in ( L~ ( Upper_Seq (C,n))) by A90;

                  then x in ( W-most ( L~ ( Cage (C,n)))) by A76, A98, EUCLID: 52, SPRECT_2: 12;

                  then (x `2 ) >= (Wmin `2 ) by PSCOMP_1: 31;

                  then (x `2 ) = (Wmin `2 ) by A97, XXREAL_0: 1;

                  then x = Wmin by A94, A98, TOPREAL3: 6;

                  then (FiP .. ( Upper_Seq (C,n))) = 1 by A46, A1, A24, A43, A100, Th37;

                  then Wmin = FiP by A1, A22, FINSEQ_5: 38;

                  hence contradiction by A16, A40, EUCLID: 52;

                end;

                  suppose x in ( LSeg (FiP, |[Ebo, (FiP `2 )]|));

                  hence contradiction by A16, A32, A40, A63, A99, TOPREAL1: 3;

                end;

              end;

              hence contradiction;

            end;

              suppose

               A101: x in ( L~ RevL);

              now

                per cases by A91, A84, A77, XBOOLE_0:def 3;

                  suppose

                   A102: x in ( L~ midU);

                  then x in (( L~ ( Upper_Seq (C,n))) /\ ( L~ ( Lower_Seq (C,n)))) by A90, A85, A101, XBOOLE_0:def 4;

                  then

                   A103: x in {Wmin, Emax} by JORDAN1E: 16;

                  now

                    per cases by A103, TARSKI:def 2;

                      suppose x = Wmin;

                      then (FiP .. ( Upper_Seq (C,n))) = 1 by A46, A1, A24, A43, A102, Th37;

                      then Wmin = FiP by A1, A22, FINSEQ_5: 38;

                      hence contradiction by A16, A40, EUCLID: 52;

                    end;

                      suppose x = Emax;

                      then (FiP .. ( Upper_Seq (C,n))) = ( len ( Upper_Seq (C,n))) by A46, A30, A24, A43, A102, Th38;

                      then Emax = FiP by A30, A22, FINSEQ_5: 38;

                      hence contradiction by A32, A40, EUCLID: 52;

                    end;

                  end;

                  hence contradiction;

                end;

                  suppose

                   A104: x in ( LSeg (FiP, |[Ebo, (FiP `2 )]|));

                  ( LSeg (FiP, |[Ebo, (FiP `2 )]|)) is horizontal by A87, SPPOL_1: 15;

                  then

                   A105: (x `2 ) = (FiP `2 ) by A104, SPPOL_1: 40;

                  consider i be Nat such that

                   A106: 1 <= i and

                   A107: (i + 1) <= ( len RevL) and

                   A108: x in ( LSeg ((RevL /. i),(RevL /. (i + 1)))) by A101, SPPOL_2: 14;

                  

                   A109: i < ( len RevL) by A107, NAT_1: 13;

                  then

                   A110: ((( Rev ( Lower_Seq (C,n))) /. i) `1 ) < sr by A21, A29, A70, A106, Th52;

                  i in ( Seg (LaP .. ( Rev ( Lower_Seq (C,n))))) by A29, A106, A109, FINSEQ_1: 1;

                  then

                   A111: (RevL /. i) = (( Rev ( Lower_Seq (C,n))) /. i) by A27, FINSEQ_5: 43;

                  (i + 1) >= 1 by NAT_1: 11;

                  then (i + 1) in ( Seg (LaP .. ( Rev ( Lower_Seq (C,n))))) by A29, A107, FINSEQ_1: 1;

                  then

                   A112: (RevL /. (i + 1)) = (( Rev ( Lower_Seq (C,n))) /. (i + 1)) by A27, FINSEQ_5: 43;

                  

                   A113: (FiP `1 ) <= (x `1 ) by A32, A40, A63, A104, TOPREAL1: 3;

                  now

                    per cases by A107, XXREAL_0: 1;

                      suppose

                       A114: (i + 1) < ( len RevL);

                      ((RevL /. i) `1 ) <= ((RevL /. (i + 1)) `1 ) or ((RevL /. (i + 1)) `1 ) <= ((RevL /. i) `1 );

                      then

                       A115: (x `1 ) <= ((RevL /. (i + 1)) `1 ) or (x `1 ) <= ((RevL /. i) `1 ) by A108, TOPREAL1: 3;

                      ((( Rev ( Lower_Seq (C,n))) /. (i + 1)) `1 ) < sr by A21, A29, A70, A114, Th52, NAT_1: 11;

                      hence contradiction by A40, A113, A111, A112, A110, A115, XXREAL_0: 2;

                    end;

                      suppose

                       A116: (i + 1) = ( len RevL);

                      then (i + 1) <= ( len ( Rev ( Lower_Seq (C,n)))) by A27, A29, FINSEQ_4: 21;

                      then ( LSeg ((( Rev ( Lower_Seq (C,n))) /. i),(( Rev ( Lower_Seq (C,n))) /. (i + 1)))) = ( LSeg (( Rev ( Lower_Seq (C,n))),i)) by A106, TOPREAL1:def 3;

                      then ( LSeg ((RevL /. i),(RevL /. (i + 1)))) is vertical or ( LSeg ((RevL /. i),(RevL /. (i + 1)))) is horizontal by A111, A112, SPPOL_1: 19;

                      hence contradiction by A44, A45, A66, A105, A108, A111, A110, A116, SPPOL_1: 16, SPPOL_1: 40;

                    end;

                  end;

                  hence contradiction;

                end;

              end;

              hence contradiction;

            end;

              suppose

               A117: x in ( LSeg (LaP, |[sr, Nbo]|));

              then

               A118: (LaP `2 ) <= (x `2 ) by A8, A86, TOPREAL1: 4;

              

               A119: (x `1 ) = (LaP `1 ) by A89, A117, SPPOL_1: 41;

              now

                per cases by A91, A84, A77, XBOOLE_0:def 3;

                  suppose x in ( L~ midU);

                  then

                  consider i be Nat such that

                   A120: 1 <= i and

                   A121: (i + 1) <= ( len midU) and

                   A122: x in ( LSeg ((midU /. i),(midU /. (i + 1)))) by SPPOL_2: 14;

                  (i + 2) >= (1 + 1) by NAT_1: 11;

                  then

                   A123: ((i + 2) - 1) >= ((1 + 1) - 1) by XREAL_1: 9;

                  i < ( len midU) by A121, NAT_1: 13;

                  then i in ( dom midU) by A120, FINSEQ_3: 25;

                  

                  then

                   A124: (midU /. i) = (( Upper_Seq (C,n)) /. ((i + 2) -' 1)) by A47, A23, A79, SPRECT_2: 3

                  .= (( Upper_Seq (C,n)) /. (i + (2 - 1))) by A123, XREAL_0:def 2;

                  ((i + 1) + 2) >= (1 + 1) by NAT_1: 11;

                  then

                   A125: (((i + 1) + 2) - 1) >= ((1 + 1) - 1) by XREAL_1: 9;

                  

                   A126: 1 <= (i + 1) by NAT_1: 11;

                  then (i + 1) in ( dom midU) by A121, FINSEQ_3: 25;

                  

                  then

                   A127: (midU /. (i + 1)) = (( Upper_Seq (C,n)) /. (((i + 1) + 2) -' 1)) by A47, A23, A79, SPRECT_2: 3

                  .= (( Upper_Seq (C,n)) /. ((i + 1) + (2 - 1))) by A125, XREAL_0:def 2;

                  

                   A128: ((i + 1) + 1) <= (((FiP .. ( Upper_Seq (C,n))) - 1) + 1) by A80, A121, XREAL_1: 7;

                  then (i + 1) < (FiP .. ( Upper_Seq (C,n))) by NAT_1: 13;

                  then

                   A129: ((midU /. i) `1 ) < sr by A21, A124, Th51, NAT_1: 11;

                  ((i + 1) + 1) <= ( len ( Upper_Seq (C,n))) by A43, A128, XXREAL_0: 2;

                  then ( LSeg ((midU /. i),(midU /. (i + 1)))) = ( LSeg (( Upper_Seq (C,n)),(i + 1))) by A124, A126, A127, TOPREAL1:def 3;

                  then

                   A130: ( LSeg ((midU /. i),(midU /. (i + 1)))) is vertical or ( LSeg ((midU /. i),(midU /. (i + 1)))) is horizontal by SPPOL_1: 19;

                  now

                    per cases by A121, XXREAL_0: 1;

                      suppose (i + 1) < ( len midU);

                      then ((i + 1) + 1) <= ( len midU) by NAT_1: 13;

                      then (((i + 1) + 1) + 1) <= (((FiP .. ( Upper_Seq (C,n))) - 1) + 1) by A80, XREAL_1: 7;

                      then ((i + 1) + 1) < (FiP .. ( Upper_Seq (C,n))) by NAT_1: 13;

                      then

                       A131: ((midU /. (i + 1)) `1 ) < sr by A21, A127, Th51, NAT_1: 11;

                      ((midU /. i) `1 ) <= ((midU /. (i + 1)) `1 ) or ((midU /. (i + 1)) `1 ) <= ((midU /. i) `1 );

                      hence contradiction by A44, A119, A122, A129, A131, TOPREAL1: 3;

                    end;

                      suppose

                       A132: (i + 1) = ( len midU);

                      then ((midU /. i) `2 ) = ((midU /. (i + 1)) `2 ) by A40, A77, A129, A130, SPPOL_1: 15, SPPOL_1: 16;

                      hence contradiction by A53, A77, A118, A122, A132, GOBOARD7: 6;

                    end;

                  end;

                  hence contradiction;

                end;

                  suppose x in ( LSeg (FiP, |[Ebo, (FiP `2 )]|));

                  hence contradiction by A53, A88, A118, SPPOL_1: 40;

                end;

              end;

              hence contradiction;

            end;

          end;

          hence contradiction;

        end;

          suppose

           A133: SW = Wmin;

          reconsider RevLS = ( Rev ( Lower_Seq (C,n))) as special FinSequence of ( TOP-REAL 2);

          set h = ((( Rev ( Lower_Seq (C,n))) -: LaP) ^ <* |[sr, Nbo]|*>);

          

           A134: <* |[sr, Nbo]|*> is one-to-one & (RevLS -: LaP) is special by FINSEQ_3: 93;

          ( rng (( Rev ( Lower_Seq (C,n))) -: LaP)) misses { |[sr, Nbo]|} by A25, ZFMISC_1: 50;

          then (( rng (( Rev ( Lower_Seq (C,n))) -: LaP)) /\ { |[sr, Nbo]|}) = {} ;

          then (( rng (( Rev ( Lower_Seq (C,n))) -: LaP)) /\ ( rng <* |[sr, Nbo]|*>)) = {} by FINSEQ_1: 38;

          then

           A135: ( rng (( Rev ( Lower_Seq (C,n))) -: LaP)) misses ( rng <* |[sr, Nbo]|*>);

          (((( Rev ( Lower_Seq (C,n))) -: LaP) /. ( len (( Rev ( Lower_Seq (C,n))) -: LaP))) `1 ) = (((( Rev ( Lower_Seq (C,n))) -: LaP) /. (LaP .. ( Rev ( Lower_Seq (C,n))))) `1 ) by A27, FINSEQ_5: 42

          .= (LaP `1 ) by A27, FINSEQ_5: 45

          .= ( |[sr, Nbo]| `1 ) by A44, EUCLID: 52

          .= (( <* |[sr, Nbo]|*> /. 1) `1 ) by FINSEQ_4: 16;

          then

          reconsider h as one-to-one special FinSequence of ( TOP-REAL 2) by A135, A134, FINSEQ_3: 91, GOBOARD2: 8;

          now

            let m be Nat;

            

             A136: ( W-bound ( L~ ( Cage (C,n)))) <= ( E-bound ( L~ ( Cage (C,n)))) by SPRECT_1: 21;

            assume m in ( dom <* |[sr, Nbo]|*>);

            then m in ( Seg 1) by FINSEQ_1: 38;

            then m = 1 by FINSEQ_1: 2, TARSKI:def 1;

            then

             A137: ( <* |[sr, Nbo]|*> /. m) = |[sr, Nbo]| by FINSEQ_4: 16;

            then (( <* |[sr, Nbo]|*> /. m) `1 ) = sr by EUCLID: 52;

            hence ( W-bound ( L~ ( Cage (C,n)))) <= (( <* |[sr, Nbo]|*> /. m) `1 ) & (( <* |[sr, Nbo]|*> /. m) `1 ) <= ( E-bound ( L~ ( Cage (C,n)))) by A136, JORDAN6: 1;

            (( <* |[sr, Nbo]|*> /. m) `2 ) = Nbo by A137, EUCLID: 52;

            hence ( S-bound ( L~ ( Cage (C,n)))) <= (( <* |[sr, Nbo]|*> /. m) `2 ) & (( <* |[sr, Nbo]|*> /. m) `2 ) <= ( N-bound ( L~ ( Cage (C,n)))) by SPRECT_1: 22;

          end;

          then

           A138: <* |[sr, Nbo]|*> is_in_the_area_of ( Cage (C,n)) by SPRECT_2:def 1;

          ( Rev ( Lower_Seq (C,n))) is_in_the_area_of ( Cage (C,n)) by JORDAN1E: 18, SPRECT_3: 51;

          then (( Rev ( Lower_Seq (C,n))) -: LaP) is_in_the_area_of ( Cage (C,n)) by A27, JORDAN1E: 1;

          then

           A139: h is_in_the_area_of ( Cage (C,n)) by A138, SPRECT_2: 24;

          

           A140: ( len h) = (( len (( Rev ( Lower_Seq (C,n))) -: LaP)) + 1) by FINSEQ_2: 16;

          

          then

           A141: ((h /. ( len h)) `2 ) = ( |[sr, Nbo]| `2 ) by FINSEQ_4: 67

          .= ( N-bound ( L~ ( Cage (C,n)))) by EUCLID: 52;

          ( L~ ( Rev ( Lower_Seq (C,n)))) = (( L~ RevL) \/ ( L~ (( Rev ( Lower_Seq (C,n))) :- LaP))) by A27, SPPOL_2: 24;

          then ( L~ RevL) c= ( L~ ( Rev ( Lower_Seq (C,n)))) by XBOOLE_1: 7;

          then

           A142: ( L~ RevL) c= ( L~ ( Lower_Seq (C,n))) by SPPOL_2: 22;

          

           A143: (LaP `2 ) <= Nbo by A20, PSCOMP_1: 24;

          (LaP .. ( Rev ( Lower_Seq (C,n)))) >= ( 0 + 1) by A28, A29, NAT_1: 13;

          then

           A144: ( len (( Rev ( Lower_Seq (C,n))) -: LaP)) >= 1 by A27, FINSEQ_5: 42;

          1 in ( dom h) by FINSEQ_5: 6;

          then (h /. 1) = (h . 1) by PARTFUN1:def 6;

          

          then ((h /. 1) `2 ) = (((( Rev ( Lower_Seq (C,n))) -: LaP) /. 1) `2 ) by A144, FINSEQ_6: 109

          .= ((( Rev ( Lower_Seq (C,n))) /. 1) `2 ) by A27, FINSEQ_5: 44

          .= ((( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) `2 ) by FINSEQ_5: 65

          .= (Wmin `2 ) by JORDAN1F: 8

          .= ( S-bound ( L~ ( Cage (C,n)))) by A133, EUCLID: 52;

          then

           A145: h is_a_v.c._for ( Cage (C,n)) by A139, A141, SPRECT_2:def 3;

          set FiP2 = ( First_Point (( L~ ( Lower_Seq (C,n))),Wmin,Emax,( Vertical_Line sr)));

          set midU = ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n)))));

          

           A146: ( |[Ebo, (FiP `2 )]| `1 ) = Ebo by EUCLID: 52;

          

           A147: ( L~ g) = (( L~ midU) \/ ( LSeg ((midU /. ( len midU)), |[Ebo, (FiP `2 )]|))) by A47, A23, SPPOL_2: 19, SPRECT_2: 7;

          

           A148: Wmin in ( rng ( Upper_Seq (C,n))) by A1, FINSEQ_6: 42;

          now

            assume (FiP .. ( Upper_Seq (C,n))) = 1;

            

            then (FiP .. ( Upper_Seq (C,n))) = ((( Upper_Seq (C,n)) /. 1) .. ( Upper_Seq (C,n))) by FINSEQ_6: 43

            .= (Wmin .. ( Upper_Seq (C,n))) by JORDAN1F: 5;

            then FiP = Wmin by A22, A148, FINSEQ_5: 9;

            hence contradiction by A16, A40, EUCLID: 52;

          end;

          then (FiP .. ( Upper_Seq (C,n))) > 1 by A24, XXREAL_0: 1;

          then

           A149: ((1 + 1) + 0 ) <= (FiP .. ( Upper_Seq (C,n))) by NAT_1: 13;

          then ((FiP .. ( Upper_Seq (C,n))) - 2) >= 0 by XREAL_1: 19;

          then ((FiP .. ( Upper_Seq (C,n))) -' 2) = ((FiP .. ( Upper_Seq (C,n))) - 2) by XREAL_0:def 2;

          

          then

           A150: ( len midU) = (((FiP .. ( Upper_Seq (C,n))) - 2) + 1) by A43, A149, JORDAN4: 8

          .= ((FiP .. ( Upper_Seq (C,n))) - (2 - 1));

          (LaP `1 ) = ( |[sr, Nbo]| `1 ) by A44, EUCLID: 52;

          then

           A151: ( LSeg (LaP, |[sr, Nbo]|)) is vertical by SPPOL_1: 16;

          ( len g) = (( len ( mid (( Upper_Seq (C,n)),2,(FiP .. ( Upper_Seq (C,n)))))) + 1) by FINSEQ_2: 16;

          then

           A152: (1 + 1) <= ( len g) by A51, XREAL_1: 7;

          

           A153: (RevL /. ( len RevL)) = (RevL /. (LaP .. ( Rev ( Lower_Seq (C,n))))) by A27, FINSEQ_5: 42

          .= LaP by A27, FINSEQ_5: 45;

          (1 + 1) <= ( len h) by A144, A140, XREAL_1: 7;

          then ( L~ g) meets ( L~ h) by A52, A145, A152, SPRECT_2: 29;

          then

          consider x be object such that

           A154: x in ( L~ g) and

           A155: x in ( L~ h) by XBOOLE_0: 3;

          reconsider x as Point of ( TOP-REAL 2) by A154;

          

           A156: ( L~ h) = (( L~ RevL) \/ ( LSeg ((RevL /. ( len RevL)), |[sr, Nbo]|))) by A27, FINSEQ_5: 47, SPPOL_2: 19;

          

           A157: (midU /. ( len midU)) = (( Upper_Seq (C,n)) /. (FiP .. ( Upper_Seq (C,n)))) by A47, A23, SPRECT_2: 9

          .= FiP by A22, FINSEQ_5: 38;

          

           A158: ( L~ midU) c= ( L~ ( Upper_Seq (C,n))) by A47, A23, SPRECT_3: 18;

          

           A159: ( L~ ( Rev ( Lower_Seq (C,n)))) = ( L~ ( Lower_Seq (C,n))) & FiP2 = LaP by A12, A17, JORDAN5C: 18, SPPOL_2: 22;

          

           A160: ( |[Ebo, (FiP `2 )]| `2 ) = (FiP `2 ) by EUCLID: 52;

          then

           A161: ( LSeg (FiP, |[Ebo, (FiP `2 )]|)) is horizontal by SPPOL_1: 15;

          now

            per cases by A155, A156, A153, XBOOLE_0:def 3;

              suppose

               A162: x in ( L~ RevL);

              now

                per cases by A154, A147, A157, XBOOLE_0:def 3;

                  suppose

                   A163: x in ( L~ midU);

                  then x in (( L~ ( Upper_Seq (C,n))) /\ ( L~ ( Lower_Seq (C,n)))) by A158, A142, A162, XBOOLE_0:def 4;

                  then

                   A164: x in {Wmin, Emax} by JORDAN1E: 16;

                  now

                    per cases by A164, TARSKI:def 2;

                      suppose x = Wmin;

                      then (FiP .. ( Upper_Seq (C,n))) = 1 by A46, A1, A24, A43, A163, Th37;

                      then Wmin = FiP by A1, A22, FINSEQ_5: 38;

                      hence contradiction by A16, A40, EUCLID: 52;

                    end;

                      suppose x = Emax;

                      then (FiP .. ( Upper_Seq (C,n))) = ( len ( Upper_Seq (C,n))) by A46, A30, A24, A43, A163, Th38;

                      then Emax = FiP by A30, A22, FINSEQ_5: 38;

                      hence contradiction by A32, A40, EUCLID: 52;

                    end;

                  end;

                  hence contradiction;

                end;

                  suppose

                   A165: x in ( LSeg (FiP, |[Ebo, (FiP `2 )]|));

                  ( LSeg (FiP, |[Ebo, (FiP `2 )]|)) is horizontal by A160, SPPOL_1: 15;

                  then

                   A166: (x `2 ) = (FiP `2 ) by A165, SPPOL_1: 40;

                  consider i be Nat such that

                   A167: 1 <= i and

                   A168: (i + 1) <= ( len RevL) and

                   A169: x in ( LSeg ((RevL /. i),(RevL /. (i + 1)))) by A162, SPPOL_2: 14;

                  

                   A170: i < ( len RevL) by A168, NAT_1: 13;

                  then

                   A171: ((( Rev ( Lower_Seq (C,n))) /. i) `1 ) < sr by A21, A29, A159, A167, Th52;

                  i in ( Seg (LaP .. ( Rev ( Lower_Seq (C,n))))) by A29, A167, A170, FINSEQ_1: 1;

                  then

                   A172: (RevL /. i) = (( Rev ( Lower_Seq (C,n))) /. i) by A27, FINSEQ_5: 43;

                  (i + 1) >= 1 by NAT_1: 11;

                  then (i + 1) in ( Seg (LaP .. ( Rev ( Lower_Seq (C,n))))) by A29, A168, FINSEQ_1: 1;

                  then

                   A173: (RevL /. (i + 1)) = (( Rev ( Lower_Seq (C,n))) /. (i + 1)) by A27, FINSEQ_5: 43;

                  

                   A174: (FiP `1 ) <= (x `1 ) by A32, A40, A146, A165, TOPREAL1: 3;

                  now

                    per cases by A168, XXREAL_0: 1;

                      suppose

                       A175: (i + 1) < ( len RevL);

                      ((RevL /. i) `1 ) <= ((RevL /. (i + 1)) `1 ) or ((RevL /. (i + 1)) `1 ) <= ((RevL /. i) `1 );

                      then

                       A176: (x `1 ) <= ((RevL /. (i + 1)) `1 ) or (x `1 ) <= ((RevL /. i) `1 ) by A169, TOPREAL1: 3;

                      ((( Rev ( Lower_Seq (C,n))) /. (i + 1)) `1 ) < sr by A21, A29, A159, A175, Th52, NAT_1: 11;

                      hence contradiction by A40, A174, A172, A173, A171, A176, XXREAL_0: 2;

                    end;

                      suppose

                       A177: (i + 1) = ( len RevL);

                      then (i + 1) <= ( len ( Rev ( Lower_Seq (C,n)))) by A27, A29, FINSEQ_4: 21;

                      then ( LSeg ((( Rev ( Lower_Seq (C,n))) /. i),(( Rev ( Lower_Seq (C,n))) /. (i + 1)))) = ( LSeg (( Rev ( Lower_Seq (C,n))),i)) by A167, TOPREAL1:def 3;

                      then ( LSeg ((RevL /. i),(RevL /. (i + 1)))) is vertical or ( LSeg ((RevL /. i),(RevL /. (i + 1)))) is horizontal by A172, A173, SPPOL_1: 19;

                      hence contradiction by A44, A45, A153, A166, A169, A172, A171, A177, SPPOL_1: 16, SPPOL_1: 40;

                    end;

                  end;

                  hence contradiction;

                end;

              end;

              hence contradiction;

            end;

              suppose

               A178: x in ( LSeg (LaP, |[sr, Nbo]|));

              then

               A179: (LaP `2 ) <= (x `2 ) by A8, A143, TOPREAL1: 4;

              

               A180: (x `1 ) = (LaP `1 ) by A151, A178, SPPOL_1: 41;

              now

                per cases by A154, A147, A157, XBOOLE_0:def 3;

                  suppose x in ( L~ midU);

                  then

                  consider i be Nat such that

                   A181: 1 <= i and

                   A182: (i + 1) <= ( len midU) and

                   A183: x in ( LSeg ((midU /. i),(midU /. (i + 1)))) by SPPOL_2: 14;

                  (i + 2) >= (1 + 1) by NAT_1: 11;

                  then

                   A184: ((i + 2) - 1) >= ((1 + 1) - 1) by XREAL_1: 9;

                  i < ( len midU) by A182, NAT_1: 13;

                  then i in ( dom midU) by A181, FINSEQ_3: 25;

                  

                  then

                   A185: (midU /. i) = (( Upper_Seq (C,n)) /. ((i + 2) -' 1)) by A47, A23, A149, SPRECT_2: 3

                  .= (( Upper_Seq (C,n)) /. (i + (2 - 1))) by A184, XREAL_0:def 2;

                  ((i + 1) + 2) >= (1 + 1) by NAT_1: 11;

                  then

                   A186: (((i + 1) + 2) - 1) >= ((1 + 1) - 1) by XREAL_1: 9;

                  

                   A187: 1 <= (i + 1) by NAT_1: 11;

                  then (i + 1) in ( dom midU) by A182, FINSEQ_3: 25;

                  

                  then

                   A188: (midU /. (i + 1)) = (( Upper_Seq (C,n)) /. (((i + 1) + 2) -' 1)) by A47, A23, A149, SPRECT_2: 3

                  .= (( Upper_Seq (C,n)) /. ((i + 1) + (2 - 1))) by A186, XREAL_0:def 2;

                  

                   A189: ((i + 1) + 1) <= (((FiP .. ( Upper_Seq (C,n))) - 1) + 1) by A150, A182, XREAL_1: 7;

                  then (i + 1) < (FiP .. ( Upper_Seq (C,n))) by NAT_1: 13;

                  then

                   A190: ((midU /. i) `1 ) < sr by A21, A185, Th51, NAT_1: 11;

                  ((i + 1) + 1) <= ( len ( Upper_Seq (C,n))) by A43, A189, XXREAL_0: 2;

                  then ( LSeg ((midU /. i),(midU /. (i + 1)))) = ( LSeg (( Upper_Seq (C,n)),(i + 1))) by A185, A187, A188, TOPREAL1:def 3;

                  then

                   A191: ( LSeg ((midU /. i),(midU /. (i + 1)))) is vertical or ( LSeg ((midU /. i),(midU /. (i + 1)))) is horizontal by SPPOL_1: 19;

                  now

                    per cases by A182, XXREAL_0: 1;

                      suppose (i + 1) < ( len midU);

                      then ((i + 1) + 1) <= ( len midU) by NAT_1: 13;

                      then (((i + 1) + 1) + 1) <= (((FiP .. ( Upper_Seq (C,n))) - 1) + 1) by A150, XREAL_1: 7;

                      then ((i + 1) + 1) < (FiP .. ( Upper_Seq (C,n))) by NAT_1: 13;

                      then

                       A192: ((midU /. (i + 1)) `1 ) < sr by A21, A188, Th51, NAT_1: 11;

                      ((midU /. i) `1 ) <= ((midU /. (i + 1)) `1 ) or ((midU /. (i + 1)) `1 ) <= ((midU /. i) `1 );

                      hence contradiction by A44, A180, A183, A190, A192, TOPREAL1: 3;

                    end;

                      suppose

                       A193: (i + 1) = ( len midU);

                      then ((midU /. i) `2 ) = ((midU /. (i + 1)) `2 ) by A40, A157, A190, A191, SPPOL_1: 15, SPPOL_1: 16;

                      hence contradiction by A53, A157, A179, A183, A193, GOBOARD7: 6;

                    end;

                  end;

                  hence contradiction;

                end;

                  suppose x in ( LSeg (FiP, |[Ebo, (FiP `2 )]|));

                  hence contradiction by A53, A161, A179, SPPOL_1: 40;

                end;

              end;

              hence contradiction;

            end;

          end;

          hence contradiction;

        end;

      end;

      hence contradiction;

    end;

    theorem :: JORDAN1G:55

    

     Th55: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for n be Nat st n > 0 holds ( L~ ( Upper_Seq (C,n))) = ( Upper_Arc ( L~ ( Cage (C,n))))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let n be Nat;

      

       A1: ( W-min ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 43;

      ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 46;

      then

       A2: ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by FINSEQ_6: 90, SPRECT_2: 43;

      

       A3: ( Upper_Seq (C,n)) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) -: ( E-max ( L~ ( Cage (C,n))))) by JORDAN1E:def 1;

      then (( Upper_Seq (C,n)) /. 1) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) /. 1) by A2, FINSEQ_5: 44;

      then

       A4: (( Upper_Seq (C,n)) /. 1) = ( W-min ( L~ ( Cage (C,n)))) by A1, FINSEQ_6: 92;

      (( Upper_Seq (C,n)) /. ( len ( Upper_Seq (C,n)))) = ((( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) -: ( E-max ( L~ ( Cage (C,n))))) /. (( E-max ( L~ ( Cage (C,n)))) .. ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))))) by A3, A2, FINSEQ_5: 42

      .= ( E-max ( L~ ( Cage (C,n)))) by A2, FINSEQ_5: 45;

      then

       A5: ( L~ ( Upper_Seq (C,n))) is_an_arc_of (( W-min ( L~ ( Cage (C,n)))),( E-max ( L~ ( Cage (C,n))))) by A4, TOPREAL1: 25;

      assume n > 0 ;

      then

       A6: (( First_Point (( L~ ( Upper_Seq (C,n))),( W-min ( L~ ( Cage (C,n)))),( E-max ( L~ ( Cage (C,n)))),( Vertical_Line ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)))) `2 ) > (( Last_Point (( L~ ( Lower_Seq (C,n))),( E-max ( L~ ( Cage (C,n)))),( W-min ( L~ ( Cage (C,n)))),( Vertical_Line ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)))) `2 ) by Th54;

      

       A7: (( Lower_Seq (C,n)) /. 1) = ((( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) :- ( E-max ( L~ ( Cage (C,n))))) /. 1) by JORDAN1E:def 2

      .= ( E-max ( L~ ( Cage (C,n)))) by FINSEQ_5: 53;

      ( Lower_Seq (C,n)) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) :- ( E-max ( L~ ( Cage (C,n))))) by JORDAN1E:def 2;

      

      then (( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) /. ( len ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))))) by A2, FINSEQ_5: 54

      .= (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) /. 1) by FINSEQ_6:def 1

      .= ( W-min ( L~ ( Cage (C,n)))) by A1, FINSEQ_6: 92;

      then

       A8: ( L~ ( Lower_Seq (C,n))) is_an_arc_of (( E-max ( L~ ( Cage (C,n)))),( W-min ( L~ ( Cage (C,n))))) by A7, TOPREAL1: 25;

      (( L~ ( Upper_Seq (C,n))) /\ ( L~ ( Lower_Seq (C,n)))) = {( W-min ( L~ ( Cage (C,n)))), ( E-max ( L~ ( Cage (C,n))))} & (( L~ ( Upper_Seq (C,n))) \/ ( L~ ( Lower_Seq (C,n)))) = ( L~ ( Cage (C,n))) by JORDAN1E: 13, JORDAN1E: 16;

      hence thesis by A5, A8, A6, JORDAN6:def 8;

    end;

    theorem :: JORDAN1G:56

    

     Th56: for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for n be Nat st n > 0 holds ( L~ ( Lower_Seq (C,n))) = ( Lower_Arc ( L~ ( Cage (C,n))))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let n be Nat;

      

       A1: ( W-min ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 43;

      

       A2: (( Lower_Seq (C,n)) /. 1) = ((( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) :- ( E-max ( L~ ( Cage (C,n))))) /. 1) by JORDAN1E:def 2

      .= ( E-max ( L~ ( Cage (C,n)))) by FINSEQ_5: 53;

      ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Cage (C,n))) by SPRECT_2: 46;

      then ( Lower_Seq (C,n)) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) :- ( E-max ( L~ ( Cage (C,n))))) & ( E-max ( L~ ( Cage (C,n)))) in ( rng ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n))))))) by FINSEQ_6: 90, JORDAN1E:def 2, SPRECT_2: 43;

      

      then (( Lower_Seq (C,n)) /. ( len ( Lower_Seq (C,n)))) = (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) /. ( len ( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))))) by FINSEQ_5: 54

      .= (( Rotate (( Cage (C,n)),( W-min ( L~ ( Cage (C,n)))))) /. 1) by FINSEQ_6:def 1

      .= ( W-min ( L~ ( Cage (C,n)))) by A1, FINSEQ_6: 92;

      then

       A3: ( L~ ( Lower_Seq (C,n))) is_an_arc_of (( E-max ( L~ ( Cage (C,n)))),( W-min ( L~ ( Cage (C,n))))) by A2, TOPREAL1: 25;

      assume n > 0 ;

      then

       A4: ( L~ ( Upper_Seq (C,n))) = ( Upper_Arc ( L~ ( Cage (C,n)))) & (( First_Point (( L~ ( Upper_Seq (C,n))),( W-min ( L~ ( Cage (C,n)))),( E-max ( L~ ( Cage (C,n)))),( Vertical_Line ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)))) `2 ) > (( Last_Point (( L~ ( Lower_Seq (C,n))),( E-max ( L~ ( Cage (C,n)))),( W-min ( L~ ( Cage (C,n)))),( Vertical_Line ((( W-bound ( L~ ( Cage (C,n)))) + ( E-bound ( L~ ( Cage (C,n))))) / 2)))) `2 ) by Th54, Th55;

      (( L~ ( Upper_Seq (C,n))) /\ ( L~ ( Lower_Seq (C,n)))) = {( W-min ( L~ ( Cage (C,n)))), ( E-max ( L~ ( Cage (C,n))))} & (( L~ ( Upper_Seq (C,n))) \/ ( L~ ( Lower_Seq (C,n)))) = ( L~ ( Cage (C,n))) by JORDAN1E: 13, JORDAN1E: 16;

      hence thesis by A3, A4, JORDAN6:def 9;

    end;

    theorem :: JORDAN1G:57

    for C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2) holds for n be Nat st n > 0 holds for i,j be Nat st 1 <= i & i <= ( len ( Gauge (C,n))) & 1 <= j & j <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (i,j)) in ( L~ ( Cage (C,n))) holds ( LSeg ((( Gauge (C,n)) * (i,1)),(( Gauge (C,n)) * (i,j)))) meets ( Lower_Arc ( L~ ( Cage (C,n))))

    proof

      let C be compact connected non vertical non horizontal Subset of ( TOP-REAL 2);

      let n be Nat;

      assume n > 0 ;

      then

       A1: ( L~ ( Lower_Seq (C,n))) = ( Lower_Arc ( L~ ( Cage (C,n)))) by Th56;

      let i,j be Nat;

      assume 1 <= i & i <= ( len ( Gauge (C,n))) & 1 <= j & j <= ( width ( Gauge (C,n))) & (( Gauge (C,n)) * (i,j)) in ( L~ ( Cage (C,n)));

      hence thesis by A1, Th46;

    end;