reserve X1, X2, Y for non empty RelStr,
  f for Function of [:X1, X2:], Y,
  x for Element of X1,
  y for Element of X2;

theorem Th20:
  for S being non empty TopSpace, T being non empty reflexive
TopSpace-like TopRelStr, x, y being Element of ContMaps (S, T) holds x <= y iff
  for i being Element of S holds [x.i, y.i] in the InternalRel of T
proof
  let S be non empty TopSpace, T be non empty reflexive TopSpace-like
  TopRelStr, x, y be Element of ContMaps (S, T);
A1: ContMaps (S, T) is full SubRelStr of T |^ the carrier of S by Def3;
  then reconsider x9 = x, y9 = y as Element of (T |^ the carrier of S) by
YELLOW_0:58;
  reconsider xa = x9, ya = y9 as Function of S, T by Th19;
  hereby
    assume
A2: x <= y;
    let i be Element of S;
    x9 <= y9 by A1,A2,YELLOW_0:59;
    then xa <= ya by WAYBEL10:11;
    then ex a, b being Element of T st a = xa.i & b = ya.i & a <= b by
YELLOW_2:def 1;
    hence [x.i, y.i] in the InternalRel of T;
  end;
  assume
A3: for i being Element of S holds [x.i, y.i] in the InternalRel of T;
  now
    let j be set;
    assume j in the carrier of S;
    then reconsider i = j as Element of S;
    reconsider a = xa.i, b = ya.i as Element of T;
    take a, b;
    thus a = xa.j & b = ya.j;
    [a, b] in the InternalRel of T by A3;
    hence a <= b;
  end;
  then xa <= ya by YELLOW_2:def 1;
  then x9 <= y9 by WAYBEL10:11;
  hence thesis by A1,YELLOW_0:60;
end;
