let n be Nat; :: thesis: for C being connected compact non horizontal non vertical Subset of ()
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 (); :: 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 ;
then 1 < (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by ;
then A3: (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) > 1 by ;
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 ;
then A8: k in dom (Cage (C,n)) by ;
then reconsider k9 = k - 1 as Nat by FINSEQ_3:26;
(S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) <= k by ;
then A9: k > 1 by ;
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 ;
A15: 1 <= i1 by ;
A16: j2 <= width (Gauge (C,n)) by ;
A17: 1 <= j2 by ;
A18: j1 <= width (Gauge (C,n)) by ;
A19: k9 + 1 < len (Cage (C,n)) by ;
A20: i1 <= len (Gauge (C,n)) by ;
((Gauge (C,n)) * (i1,j1)) `2 = S-bound (L~ (Cage (C,n))) by
.= ((Gauge (C,n)) * (i1,1)) `2 by ;
then A21: j1 <= 1 by ;
k >= 1 + 1 by ;
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 ;
A28: i1 = i4 by ;
A29: j1 = j4 by ;
A30: 1 <= i3 by ;
A31: i3 <= len (Gauge (C,n)) by ;
A32: 1 <= j1 by ;
then A33: j1 = 1 by ;
A34: i3 = i4
proof
assume A35: i3 <> i4 ; :: thesis: contradiction
per cases ( ( j3 = j4 & i3 + 1 = i4 ) or ( j3 = j4 & i3 = i4 + 1 ) ) by ;
suppose ( j3 = j4 & i3 + 1 = i4 ) ; :: thesis: contradiction
end;
suppose A36: ( j3 = j4 & i3 = i4 + 1 ) ; :: thesis: contradiction
k9 < len (Cage (C,n)) by ;
then (Gauge (C,n)) * (i3,j3) in S-most (L~ (Cage (C,n))) by ;
then A37: ((Gauge (C,n)) * ((i4 + 1),j4)) `1 <= ((Gauge (C,n)) * (i4,j4)) `1 by ;
i4 < i4 + 1 by NAT_1:13;
hence contradiction by A15, A32, A18, A28, A29, A31, A36, A37, GOBOARD5:3; :: thesis: verum
end;
end;
end;
A38: ( 1 <= i2 & i2 <= len (Gauge (C,n)) ) by ;
A39: k9 + 1 = k ;
A40: 1 <= j3 by ;
j1 = j2
proof
assume A41: j1 <> j2 ; :: thesis: contradiction
per cases ( ( j1 = j2 + 1 & i1 = i2 ) or ( j1 + 1 = j2 & i1 = i2 ) ) by ;
suppose ( j1 = j2 + 1 & i1 = i2 ) ; :: thesis: contradiction
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 ;
( (Cage (C,n)) /. k9 in LSeg ((Cage (C,n)),k9) & (Cage (C,n)) /. (k + 1) in LSeg ((Cage (C,n)),k) ) by ;
then (Cage (C,n)) /. (k + 1) in {((Cage (C,n)) /. k)} by ;
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;
end;
end;
then ((Gauge (C,n)) * (i1,j1)) `2 = ((Gauge (C,n)) * (1,j2)) `2 by
.= ((Gauge (C,n)) * (i2,j2)) `2 by ;
hence ((Cage (C,n)) /. (k + 1)) `2 = S-bound (L~ (Cage (C,n))) by ; :: thesis: verum