reserve n for Nat;

theorem Th57:
  for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j,k be Nat st 1 < i & i < len Gauge(C,n) & 1 <= j &
k <= width Gauge(C,n) & Gauge(C,n)*(i,k) in L~Upper_Seq(C,n) & Gauge(C,n)*(i,j)
  in L~Lower_Seq(C,n) holds j <> k
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  let i,j,k be Nat;
  assume that
A1: 1 < i and
A2: i < len Gauge(C,n) and
A3: 1 <= j and
A4: k <= width Gauge(C,n) and
A5: Gauge(C,n)*(i,k) in L~Upper_Seq(C,n) and
A6: Gauge(C,n)*(i,j) in L~Lower_Seq(C,n) and
A7: j = k;
A8: [i,j] in Indices Gauge(C,n) by A1,A2,A3,A4,A7,MATRIX_0:30;
  Gauge(C,n)*(i,k) in L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n) by A5,A6,A7,
XBOOLE_0:def 4;
  then
A9: Gauge(C,n)*(i,k) in {W-min L~Cage(C,n),E-max L~Cage(C,n)} by JORDAN1E:16;
A10: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
  len Gauge(C,n) >= 4 by JORDAN8:10;
  then
A11: len Gauge(C,n) >= 1 by XXREAL_0:2;
  then
A12: [len Gauge(C,n),j] in Indices Gauge(C,n) by A3,A4,A7,MATRIX_0:30;
A13: [1,j] in Indices Gauge(C,n) by A3,A4,A7,A11,MATRIX_0:30;
  per cases by A9,TARSKI:def 2;
  suppose
A14: Gauge(C,n)*(i,k) = W-min L~Cage(C,n);
    Gauge(C,n)*(1,j)`1 = W-bound L~Cage(C,n) by A3,A4,A7,A10,JORDAN1A:73;
    then
(W-min L~Cage(C,n))`1 <> W-bound L~Cage(C,n) by A1,A7,A8,A13,A14,JORDAN1G:7;
    hence contradiction by EUCLID:52;
  end;
  suppose
A15: Gauge(C,n)*(i,k) = E-max L~Cage(C,n);
    Gauge(C,n)*(len Gauge(C,n),j)`1 = E-bound L~Cage(C,n) by A3,A4,A7,A10,
JORDAN1A:71;
    then
(E-max L~Cage(C,n))`1 <> E-bound L~Cage(C,n) by A2,A7,A8,A12,A15,JORDAN1G:7;
    hence contradiction by EUCLID:52;
  end;
end;
