reserve n for Nat;

theorem
  for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for k be 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)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
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;
  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);
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 be 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 be 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
    assume
A35: i3 <> i4;
    per cases by A27,A35;
    suppose
      j3 = j4 & i3+1 = i4;
      hence contradiction by A7,A21,A22,A23,A24,A25,A26,A29,JORDAN10:3;
    end;
    suppose
A36:  j3 = j4 & i3 = i4+1;
      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;
    end;
  end;
A38: 1 <= i2 & i2 <= len Gauge(C,n) by A12,MATRIX_0:32;
A39: k9+1 = k;
A40: 1 <= j3 by A23,MATRIX_0:32;
  j1 = j2
  proof
    assume
A41: j1 <> j2;
    per cases by A14,A41;
    suppose
      j1 = j2+1 & i1 = i2;
      hence contradiction by A17,A21,NAT_1:13;
    end;
    suppose
A42:  j1+1 = j2 & i1 = i2;
      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;
    end;
  end;
  then Gauge(C,n)*(i1,j1)`2 = Gauge(C,n)*(1,j2)`2 by A15,A20,A32,A18,GOBOARD5:1
    .= Gauge(C,n)*(i2,j2)`2 by A38,A17,A16,GOBOARD5:1;
  hence thesis by A6,A11,A13,EUCLID:52;
end;
