reserve i, i1, i2, j, j1, j2, k, m, n, t for Nat,
  D for non empty Subset of TOP-REAL 2,
  E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p, q, x for Point of TOP-REAL 2,
  r, s for Real;

theorem
  1 <= i & i <= len Gauge(E,n) & 1 <= j & j <= len Gauge(E,m) & m <= n
  implies Gauge(E,m)*(j,1)`2 <= Gauge(E,n)*(i,1)`2
proof
  set a = N-bound E, s = S-bound E, G = Gauge(E,n), M = Gauge(E,m);
  assume that
A1: 1 <= i & i <= len G and
A2: 1 <= j & j <= len M and
A3: m <= n;
A4: len M = width M by JORDAN8:def 1;
  1 <= len M by A2,XXREAL_0:2;
  then [j,1] in Indices M by A2,A4,MATRIX_0:30;
  then
A5: M*(j,1)`2 = s-(a-s)/(2|^m) by Lm11;
A6: len G = width G by JORDAN8:def 1;
  1 <= len G by A1,XXREAL_0:2;
  then [i,1] in Indices G by A1,A6,MATRIX_0:30;
  then 0 < a - s & G*(i,1)`2 = s-(a-s)/(2|^n) by Lm11,SPRECT_1:32,XREAL_1:50;
  hence thesis by A3,A5,Lm9;
end;
