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 Th1:
  1 <= k & k+1 <= len Cage(C,n) & [i,j] in Indices Gauge(C,n) & [i,
j+1] in Indices Gauge(C,n) & Cage(C,n)/.k = Gauge(C,n)*(i,j) & Cage(C,n)/.(k+1)
  = Gauge(C,n)*(i,j+1) implies i < len Gauge(C,n)
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,j+1] in Indices Gauge(C,n) & Cage(C,n)/.k = Gauge(C,n)*(i,j) &
  Cage(C,n)/.(k+1) = Gauge(C,n)*(i,j+1);
  assume
A4: i >= len G;
  len G = width G by JORDAN8:def 1;
  then
A5: j <= len G by A2,MATRIX_0:32;
  i <= len G by A2,MATRIX_0:32;
  then
A6: i = len G by A4,XXREAL_0:1;
  f is_sequence_on G by JORDAN9:def 1;
  then right_cell(f,k,G) = cell(G,i,j) by A1,A2,A3,GOBRD13:22;
  hence contradiction by A1,A6,A5,JORDAN8:16,JORDAN9:31;
end;
