theorem Th47:
  for X, Y being complex-membered set
  holds X c= Y implies multRel(X,z) c= multRel(Y,z)
proof
  let X, Y be complex-membered set;
  assume A1: X c= Y;
  now
    let x,y be object;
    reconsider a=x,b=y as set by TARSKI:1;
    assume A2: [x,y] in multRel(X,z);
    then [a,b] in multRel(X,z);
    then a in X & b in X by MMLQUER2:4;
    then reconsider a,b as Complex;
    [a,b] in multRel(X,z) by A2;
    then a in X & b in X & b = z * a by Th42;
    hence [x,y] in multRel(Y,z) by A1, Th42;
  end;
  hence thesis by RELAT_1:def 3;
end;
