theorem Th39:
  for P,R being Relation st rng R c= rng P & P is positive-yielding holds
  R is positive-yielding
  proof
    let P,R be Relation;
    assume
A1: rng R c= rng P & P is positive-yielding;
    let r be Real;
    thus thesis by A1,PARTFUN3:def 1;
  end;
