theorem Th155:
  r--F c= r--G implies F c= G
proof
  assume
A1: r--F c= r--G;
  let i;
  assume i in F;
  then r-i in r--F by Th152;
  then ex w st r-i = r-w & w in G by A1,Th154;
  hence thesis by XXREAL_3:12;
end;
