let n be Nat; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)

for k being Nat st 1 <= k & k + 1 <= len (Cage (C,n)) & (Cage (C,n)) /. k = S-max (L~ (Cage (C,n))) holds

((Cage (C,n)) /. (k + 1)) `2 = S-bound (L~ (Cage (C,n)))

let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for k being Nat st 1 <= k & k + 1 <= len (Cage (C,n)) & (Cage (C,n)) /. k = S-max (L~ (Cage (C,n))) holds

((Cage (C,n)) /. (k + 1)) `2 = S-bound (L~ (Cage (C,n)))

A1: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def 1;

A2: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32;

then 1 < (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:69;

then 1 < (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:70, XXREAL_0:2;

then 1 < (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:71, XXREAL_0:2;

then A3: (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) > 1 by A2, SPRECT_2:72, XXREAL_0:2;

let k be Nat; :: thesis: ( 1 <= k & k + 1 <= len (Cage (C,n)) & (Cage (C,n)) /. k = S-max (L~ (Cage (C,n))) implies ((Cage (C,n)) /. (k + 1)) `2 = S-bound (L~ (Cage (C,n))) )

assume that

A4: 1 <= k and

A5: k + 1 <= len (Cage (C,n)) and

A6: (Cage (C,n)) /. k = S-max (L~ (Cage (C,n))) ; :: thesis: ((Cage (C,n)) /. (k + 1)) `2 = S-bound (L~ (Cage (C,n)))

A7: k < len (Cage (C,n)) by A5, NAT_1:13;

then A8: k in dom (Cage (C,n)) by A4, FINSEQ_3:25;

then reconsider k9 = k - 1 as Nat by FINSEQ_3:26;

(S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) <= k by A6, A8, FINSEQ_5:39;

then A9: k > 1 by A3, XXREAL_0:2;

then consider i1, j1, i2, j2 being Nat such that

A10: [i1,j1] in Indices (Gauge (C,n)) and

A11: (Cage (C,n)) /. k = (Gauge (C,n)) * (i1,j1) and

A12: [i2,j2] in Indices (Gauge (C,n)) and

A13: (Cage (C,n)) /. (k + 1) = (Gauge (C,n)) * (i2,j2) and

A14: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A1, A5, JORDAN8:3;

A15: 1 <= i1 by A10, MATRIX_0:32;

A16: j2 <= width (Gauge (C,n)) by A12, MATRIX_0:32;

A17: 1 <= j2 by A12, MATRIX_0:32;

A18: j1 <= width (Gauge (C,n)) by A10, MATRIX_0:32;

A19: k9 + 1 < len (Cage (C,n)) by A5, NAT_1:13;

A20: i1 <= len (Gauge (C,n)) by A10, MATRIX_0:32;

((Gauge (C,n)) * (i1,j1)) `2 = S-bound (L~ (Cage (C,n))) by A6, A11, EUCLID:52

.= ((Gauge (C,n)) * (i1,1)) `2 by A15, A20, JORDAN1A:72 ;

then A21: j1 <= 1 by A15, A20, A18, GOBOARD5:4;

k >= 1 + 1 by A9, NAT_1:13;

then A22: k9 >= (1 + 1) - 1 by XREAL_1:9;

then consider i3, j3, i4, j4 being Nat such that

A23: [i3,j3] in Indices (Gauge (C,n)) and

A24: (Cage (C,n)) /. k9 = (Gauge (C,n)) * (i3,j3) and

A25: [i4,j4] in Indices (Gauge (C,n)) and

A26: (Cage (C,n)) /. (k9 + 1) = (Gauge (C,n)) * (i4,j4) and

A27: ( ( i3 = i4 & j3 + 1 = j4 ) or ( i3 + 1 = i4 & j3 = j4 ) or ( i3 = i4 + 1 & j3 = j4 ) or ( i3 = i4 & j3 = j4 + 1 ) ) by A1, A7, JORDAN8:3;

A28: i1 = i4 by A10, A11, A25, A26, GOBOARD1:5;

A29: j1 = j4 by A10, A11, A25, A26, GOBOARD1:5;

A30: 1 <= i3 by A23, MATRIX_0:32;

A31: i3 <= len (Gauge (C,n)) by A23, MATRIX_0:32;

A32: 1 <= j1 by A10, MATRIX_0:32;

then A33: j1 = 1 by A21, XXREAL_0:1;

A34: i3 = i4

A39: k9 + 1 = k ;

A40: 1 <= j3 by A23, MATRIX_0:32;

j1 = j2

.= ((Gauge (C,n)) * (i2,j2)) `2 by A38, A17, A16, GOBOARD5:1 ;

hence ((Cage (C,n)) /. (k + 1)) `2 = S-bound (L~ (Cage (C,n))) by A6, A11, A13, EUCLID:52; :: thesis: verum

for k being Nat st 1 <= k & k + 1 <= len (Cage (C,n)) & (Cage (C,n)) /. k = S-max (L~ (Cage (C,n))) holds

((Cage (C,n)) /. (k + 1)) `2 = S-bound (L~ (Cage (C,n)))

let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for k being Nat st 1 <= k & k + 1 <= len (Cage (C,n)) & (Cage (C,n)) /. k = S-max (L~ (Cage (C,n))) holds

((Cage (C,n)) /. (k + 1)) `2 = S-bound (L~ (Cage (C,n)))

A1: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def 1;

A2: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32;

then 1 < (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:69;

then 1 < (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:70, XXREAL_0:2;

then 1 < (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:71, XXREAL_0:2;

then A3: (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) > 1 by A2, SPRECT_2:72, XXREAL_0:2;

let k be Nat; :: thesis: ( 1 <= k & k + 1 <= len (Cage (C,n)) & (Cage (C,n)) /. k = S-max (L~ (Cage (C,n))) implies ((Cage (C,n)) /. (k + 1)) `2 = S-bound (L~ (Cage (C,n))) )

assume that

A4: 1 <= k and

A5: k + 1 <= len (Cage (C,n)) and

A6: (Cage (C,n)) /. k = S-max (L~ (Cage (C,n))) ; :: thesis: ((Cage (C,n)) /. (k + 1)) `2 = S-bound (L~ (Cage (C,n)))

A7: k < len (Cage (C,n)) by A5, NAT_1:13;

then A8: k in dom (Cage (C,n)) by A4, FINSEQ_3:25;

then reconsider k9 = k - 1 as Nat by FINSEQ_3:26;

(S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) <= k by A6, A8, FINSEQ_5:39;

then A9: k > 1 by A3, XXREAL_0:2;

then consider i1, j1, i2, j2 being Nat such that

A10: [i1,j1] in Indices (Gauge (C,n)) and

A11: (Cage (C,n)) /. k = (Gauge (C,n)) * (i1,j1) and

A12: [i2,j2] in Indices (Gauge (C,n)) and

A13: (Cage (C,n)) /. (k + 1) = (Gauge (C,n)) * (i2,j2) and

A14: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A1, A5, JORDAN8:3;

A15: 1 <= i1 by A10, MATRIX_0:32;

A16: j2 <= width (Gauge (C,n)) by A12, MATRIX_0:32;

A17: 1 <= j2 by A12, MATRIX_0:32;

A18: j1 <= width (Gauge (C,n)) by A10, MATRIX_0:32;

A19: k9 + 1 < len (Cage (C,n)) by A5, NAT_1:13;

A20: i1 <= len (Gauge (C,n)) by A10, MATRIX_0:32;

((Gauge (C,n)) * (i1,j1)) `2 = S-bound (L~ (Cage (C,n))) by A6, A11, EUCLID:52

.= ((Gauge (C,n)) * (i1,1)) `2 by A15, A20, JORDAN1A:72 ;

then A21: j1 <= 1 by A15, A20, A18, GOBOARD5:4;

k >= 1 + 1 by A9, NAT_1:13;

then A22: k9 >= (1 + 1) - 1 by XREAL_1:9;

then consider i3, j3, i4, j4 being Nat such that

A23: [i3,j3] in Indices (Gauge (C,n)) and

A24: (Cage (C,n)) /. k9 = (Gauge (C,n)) * (i3,j3) and

A25: [i4,j4] in Indices (Gauge (C,n)) and

A26: (Cage (C,n)) /. (k9 + 1) = (Gauge (C,n)) * (i4,j4) and

A27: ( ( i3 = i4 & j3 + 1 = j4 ) or ( i3 + 1 = i4 & j3 = j4 ) or ( i3 = i4 + 1 & j3 = j4 ) or ( i3 = i4 & j3 = j4 + 1 ) ) by A1, A7, JORDAN8:3;

A28: i1 = i4 by A10, A11, A25, A26, GOBOARD1:5;

A29: j1 = j4 by A10, A11, A25, A26, GOBOARD1:5;

A30: 1 <= i3 by A23, MATRIX_0:32;

A31: i3 <= len (Gauge (C,n)) by A23, MATRIX_0:32;

A32: 1 <= j1 by A10, MATRIX_0:32;

then A33: j1 = 1 by A21, XXREAL_0:1;

A34: i3 = i4

proof

A38:
( 1 <= i2 & i2 <= len (Gauge (C,n)) )
by A12, MATRIX_0:32;
assume A35:
i3 <> i4
; :: thesis: contradiction

end;per cases
( ( j3 = j4 & i3 + 1 = i4 ) or ( j3 = j4 & i3 = i4 + 1 ) )
by A27, A35;

end;

suppose A36:
( j3 = j4 & i3 = i4 + 1 )
; :: thesis: contradiction

k9 < len (Cage (C,n))
by A19, NAT_1:13;

then (Gauge (C,n)) * (i3,j3) in S-most (L~ (Cage (C,n))) by A33, A22, A24, A29, A30, A31, A36, JORDAN1A:60;

then A37: ((Gauge (C,n)) * ((i4 + 1),j4)) `1 <= ((Gauge (C,n)) * (i4,j4)) `1 by A6, A26, A36, PSCOMP_1:55;

i4 < i4 + 1 by NAT_1:13;

hence contradiction by A15, A32, A18, A28, A29, A31, A36, A37, GOBOARD5:3; :: thesis: verum

end;then (Gauge (C,n)) * (i3,j3) in S-most (L~ (Cage (C,n))) by A33, A22, A24, A29, A30, A31, A36, JORDAN1A:60;

then A37: ((Gauge (C,n)) * ((i4 + 1),j4)) `1 <= ((Gauge (C,n)) * (i4,j4)) `1 by A6, A26, A36, PSCOMP_1:55;

i4 < i4 + 1 by NAT_1:13;

hence contradiction by A15, A32, A18, A28, A29, A31, A36, A37, GOBOARD5:3; :: thesis: verum

A39: k9 + 1 = k ;

A40: 1 <= j3 by A23, MATRIX_0:32;

j1 = j2

proof

then ((Gauge (C,n)) * (i1,j1)) `2 =
((Gauge (C,n)) * (1,j2)) `2
by A15, A20, A32, A18, GOBOARD5:1
assume A41:
j1 <> j2
; :: thesis: contradiction

end;per cases
( ( j1 = j2 + 1 & i1 = i2 ) or ( j1 + 1 = j2 & i1 = i2 ) )
by A14, A41;

end;

suppose A42:
( j1 + 1 = j2 & i1 = i2 )
; :: thesis: contradiction

k9 + (1 + 1) <= len (Cage (C,n))
by A5;

then A43: (LSeg ((Cage (C,n)),k9)) /\ (LSeg ((Cage (C,n)),k)) = {((Cage (C,n)) /. k)} by A22, A39, TOPREAL1:def 6;

( (Cage (C,n)) /. k9 in LSeg ((Cage (C,n)),k9) & (Cage (C,n)) /. (k + 1) in LSeg ((Cage (C,n)),k) ) by A4, A5, A7, A22, A39, TOPREAL1:21;

then (Cage (C,n)) /. (k + 1) in {((Cage (C,n)) /. k)} by A13, A21, A24, A27, A28, A29, A40, A34, A42, A43, NAT_1:13, XBOOLE_0:def 4;

then (Cage (C,n)) /. (k + 1) = (Cage (C,n)) /. k by TARSKI:def 1;

hence contradiction by A10, A11, A12, A13, A41, GOBOARD1:5; :: thesis: verum

end;then A43: (LSeg ((Cage (C,n)),k9)) /\ (LSeg ((Cage (C,n)),k)) = {((Cage (C,n)) /. k)} by A22, A39, TOPREAL1:def 6;

( (Cage (C,n)) /. k9 in LSeg ((Cage (C,n)),k9) & (Cage (C,n)) /. (k + 1) in LSeg ((Cage (C,n)),k) ) by A4, A5, A7, A22, A39, TOPREAL1:21;

then (Cage (C,n)) /. (k + 1) in {((Cage (C,n)) /. k)} by A13, A21, A24, A27, A28, A29, A40, A34, A42, A43, NAT_1:13, XBOOLE_0:def 4;

then (Cage (C,n)) /. (k + 1) = (Cage (C,n)) /. k by TARSKI:def 1;

hence contradiction by A10, A11, A12, A13, A41, GOBOARD1:5; :: thesis: verum

.= ((Gauge (C,n)) * (i2,j2)) `2 by A38, A17, A16, GOBOARD5:1 ;

hence ((Cage (C,n)) /. (k + 1)) `2 = S-bound (L~ (Cage (C,n))) by A6, A11, A13, EUCLID:52; :: thesis: verum