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 = E-max L~Cage(C,n) holds (Cage(C,n)/.(k+1))`1 = E-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: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
A3: 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
A4: (E-max L~Cage(C,n))..Cage(C,n) > 1 by A3,SPRECT_2:70,XXREAL_0:2;
  let k be Nat;
  assume that
A5: 1 <= k and
A6: k+1 <= len Cage(C,n) and
A7: Cage(C,n)/.k = E-max L~Cage(C,n);
A8: k < len Cage(C,n) by A6,NAT_1:13;
  then
A9: k in dom Cage(C,n) by A5,FINSEQ_3:25;
  then reconsider k9=k-1 as Nat by FINSEQ_3:26;
  (E-max L~Cage(C,n))..Cage(C,n) <= k by A7,A9,FINSEQ_5:39;
  then
A10: k > 1 by A4,XXREAL_0:2;
  then consider i1,j1,i2,j2 be Nat such that
A11: [i1,j1] in Indices Gauge(C,n) and
A12: Cage(C,n)/.k = Gauge(C,n)*(i1,j1) and
A13: [i2,j2] in Indices Gauge(C,n) and
A14: Cage(C,n)/.(k+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 A1,A6,JORDAN8:3;
A16: 1 <= i1 by A11,MATRIX_0:32;
A17: k9+1 < len Cage(C,n) by A6,NAT_1:13;
A18: 1 <= j1 by A11,MATRIX_0:32;
A19: j2 <= width Gauge(C,n) by A13,MATRIX_0:32;
A20: i2 <= len Gauge(C,n) by A13,MATRIX_0:32;
A21: j1 <= width Gauge(C,n) by A11,MATRIX_0:32;
  Gauge(C,n)*(i1,j1)`1 = E-bound L~Cage(C,n) by A7,A12,EUCLID:52
    .= Gauge(C,n)*(len Gauge(C,n),j1)`1 by A18,A21,A2,JORDAN1A:71;
  then
A22: i1 >= len Gauge(C,n) by A16,A18,A21,GOBOARD5:3;
  k >= 1+1 by A10,NAT_1:13;
  then
A23: k9 >= 1+1-1 by XREAL_1:9;
  then consider i3,j3,i4,j4 be Nat such that
A24: [i3,j3] in Indices Gauge(C,n) and
A25: Cage(C,n)/.k9 = Gauge(C,n)*(i3,j3) and
A26: [i4,j4] in Indices Gauge(C,n) and
A27: Cage(C,n)/.(k9+1) = Gauge(C,n)*(i4,j4) and
A28: 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,A8,JORDAN8:3;
A29: i1 = i4 by A11,A12,A26,A27,GOBOARD1:5;
A30: j1 = j4 by A11,A12,A26,A27,GOBOARD1:5;
A31: 1 <= j3 by A24,MATRIX_0:32;
A32: j3 <= width Gauge(C,n) by A24,MATRIX_0:32;
A33: i1 <= len Gauge(C,n) by A11,MATRIX_0:32;
  then
A34: i1 = len Gauge(C,n) by A22,XXREAL_0:1;
A35: j3 = j4
  proof
    assume
A36: j3 <> j4;
    per cases by A28,A36;
    suppose
      i3 = i4 & j3+1 = j4;
      hence contradiction by A8,A22,A23,A24,A25,A26,A27,A29,JORDAN10:1;
    end;
    suppose
A37:  i3 = i4 & j3 = j4+1;
      k9 < len Cage(C,n) by A17,NAT_1:13;
      then
      Gauge(C,n)*(i3,j3) in E-most L~Cage(C,n) by A34,A23,A25,A29,A31,A32,A37,
JORDAN1A:61;
      then
A38:  (Gauge(C,n)*(i4,j4+1))`2 <= (Gauge(C,n)*(i4,j4))`2 by A7,A27,A37,
PSCOMP_1:47;
      j4 < j4+1 by NAT_1:13;
      hence contradiction by A16,A33,A18,A29,A30,A32,A37,A38,GOBOARD5:4;
    end;
  end;
A39: 1 <= i2 & 1 <= j2 by A13,MATRIX_0:32;
A40: k9+1 = k;
A41: i3 <= len Gauge(C,n) by A24,MATRIX_0:32;
  i1 = i2
  proof
    assume
A42: i1 <> i2;
    per cases by A15,A42;
    suppose
      i1+1 = i2 & j1 = j2;
      hence contradiction by A20,A22,NAT_1:13;
    end;
    suppose
A43:  i1 = i2+1 & j1 = j2;
      k9+(1+1) <= len Cage(C,n) by A6;
      then
A44:  LSeg(Cage(C,n),k9) /\ LSeg(Cage(C,n),k) = {Cage(C,n)/.k} by A23,A40,
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 A5,A6,A8,A23,A40,TOPREAL1:21;
      then
      Cage(C,n)/.(k+1) in {Cage(C,n)/.k} by A14,A22,A25,A28,A29,A30,A41,A35,A43
,A44,NAT_1:13,XBOOLE_0:def 4;
      then Cage(C,n)/.(k+1) = Cage(C,n)/.k by TARSKI:def 1;
      hence contradiction by A11,A12,A13,A14,A42,GOBOARD1:5;
    end;
  end;
  then Gauge(C,n)*(i1,j1)`1 = Gauge(C,n)*(i2,1)`1 by A16,A33,A18,A21,GOBOARD5:2
    .= Gauge(C,n)*(i2,j2)`1 by A20,A39,A19,GOBOARD5:2;
  hence thesis by A7,A12,A14,EUCLID:52;
end;
