reserve i, j, k, n for Nat,
  P for Subset of TOP-REAL 2,
  C for connected compact non vertical non horizontal Subset of TOP-REAL 2;

theorem Th3:
  1 <= k & k+1 <= len Cage(C,n) & [i,j] in Indices Gauge(C,n) & [i+
1,j] in Indices Gauge(C,n) & Cage(C,n)/.k = Gauge(C,n)*(i,j) & Cage(C,n)/.(k+1)
  = Gauge(C,n)*(i+1,j) implies j > 1
proof
  set f = Cage(C,n), G = Gauge(C,n);
  assume that
A1: 1 <= k & k+1 <= len Cage(C,n) and
A2: [i,j] in Indices Gauge(C,n) and
A3: [i+1,j] in Indices Gauge(C,n) & Cage(C,n)/.k = Gauge(C,n)*(i,j) &
  Cage(C,n)/.(k+1) = Gauge(C,n)*(i+1,j);
  assume
A4: j <= 1;
  1 <= j by A2,MATRIX_0:32;
  then j = 1 by A4,XXREAL_0:1;
  then
A5: j-'1 = 0 by XREAL_1:232;
A6: i <= len G by A2,MATRIX_0:32;
  f is_sequence_on G by JORDAN9:def 1;
  then right_cell(f,k,G) = cell(G,i,j-'1) by A1,A2,A3,GOBRD13:24;
  hence contradiction by A1,A5,A6,JORDAN8:17,JORDAN9:31;
end;
