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 = W-min L~Cage(C,n) holds (Cage(C,n)/.(k+1))`1 = W-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 1 < (S-max L~Cage(C,n))..Cage(C,n) by A2,SPRECT_2:72,XXREAL_0:2;
  then 1 < (S-min L~Cage(C,n))..Cage(C,n) by A2,SPRECT_2:73,XXREAL_0:2;
  then
A3: (W-min L~Cage(C,n))..Cage(C,n) > 1 by A2,SPRECT_2:74,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 = W-min 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;
  (W-min 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: i1 <= len Gauge(C,n) by A10,MATRIX_0:32;
A16: j2 <= width Gauge(C,n) by A12,MATRIX_0:32;
A17: 1 <= i2 by A12,MATRIX_0:32;
A18: j1 <= width Gauge(C,n) by A10,MATRIX_0:32;
A19: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
A20: i2 <= len Gauge(C,n) & 1 <= j2 by A12,MATRIX_0:32;
A21: k9+1 = k;
A22: k9+1 < len Cage(C,n) by A5,NAT_1:13;
A23: 1 <= j1 by A10,MATRIX_0:32;
  Gauge(C,n)*(i1,j1)`1 = W-bound L~Cage(C,n) by A6,A11,EUCLID:52
    .= Gauge(C,n)*(1,j1)`1 by A23,A18,A19,JORDAN1A:73;
  then
A24: i1 <= 1 by A15,A23,A18,GOBOARD5:3;
  k >= 1+1 by A9,NAT_1:13;
  then
A25: k9 >= 1+1-1 by XREAL_1:9;
  then consider i3,j3,i4,j4 be Nat such that
A26: [i3,j3] in Indices Gauge(C,n) and
A27: Cage(C,n)/.k9 = Gauge(C,n)*(i3,j3) and
A28: [i4,j4] in Indices Gauge(C,n) and
A29: Cage(C,n)/.(k9+1) = Gauge(C,n)*(i4,j4) and
A30: 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;
A31: i1 = i4 by A10,A11,A28,A29,GOBOARD1:5;
A32: j1 = j4 by A10,A11,A28,A29,GOBOARD1:5;
A33: j3 <= width Gauge(C,n) by A26,MATRIX_0:32;
A34: 1 <= j3 by A26,MATRIX_0:32;
A35: 1 <= i1 by A10,MATRIX_0:32;
  then
A36: i1 = 1 by A24,XXREAL_0:1;
A37: j3 = j4
  proof
    assume
A38: j3 <> j4;
    per cases by A30,A38;
    suppose
      i3 = i4 & j3 = j4+1;
      hence contradiction by A7,A24,A25,A26,A27,A28,A29,A31,JORDAN10:2;
    end;
    suppose
A39:  i3 = i4 & j3+1 = j4;
      k9 < len Cage(C,n) by A22,NAT_1:13;
      then
      Gauge(C,n)*(i3,j3) in W-most L~Cage(C,n) by A36,A25,A27,A31,A34,A33,A39,
JORDAN1A:59;
      then
A40:  (Gauge(C,n)*(i3,j3+1))`2 <= (Gauge(C,n)*(i3,j3))`2 by A6,A29,A39,
PSCOMP_1:31;
      j3 < j3+1 by NAT_1:13;
      hence contradiction by A35,A15,A18,A31,A32,A34,A39,A40,GOBOARD5:4;
    end;
  end;
A41: 1 <= i3 by A26,MATRIX_0:32;
  i1 = i2
  proof
    assume
A42: i1 <> i2;
    per cases by A14,A42;
    suppose
      i1 = i2+1 & j1 = j2;
      hence contradiction by A17,A24,NAT_1:13;
    end;
    suppose
A43:  i1+1 = i2 & j1 = j2;
      k9+(1+1) <= len Cage(C,n) by A5;
      then
A44:  LSeg(Cage(C,n),k9) /\ LSeg(Cage(C,n),k) = {Cage(C,n)/.k} by A25,A21,
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,A25,A21,TOPREAL1:21;
      then Cage(C,n)/.(k+1) in {Cage(C,n)/.k} by A13,A24,A27,A30,A31,A32,A41
,A37,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 A10,A11,A12,A13,A42,GOBOARD1:5;
    end;
  end;
  then Gauge(C,n)*(i1,j1)`1 = Gauge(C,n)*(i2,1)`1 by A35,A15,A23,A18,GOBOARD5:2
    .= Gauge(C,n)*(i2,j2)`1 by A17,A20,A16,GOBOARD5:2;
  hence thesis by A6,A11,A13,EUCLID:52;
end;
