reserve x,y for object,X for set,
  f for Function,
  R,S for Relation;

theorem Th3:
  for S being real-valued Relation st R c= S holds R is real-valued
proof
  let S be real-valued Relation;
  assume R c= S;
  then
A1: rng R c= rng S by RELAT_1:11;
  rng S c= REAL by Def3;
  hence rng R c= REAL by A1;
end;
