reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem Th119:
  rng(P*R) = R.:(rng P)
proof
  for z being object holds z in rng(P*R) iff z in R.:(rng P)
  proof let z be object;
    thus z in rng(P*R) implies z in R.:(rng P)
    proof
      assume z in rng(P*R);
      then consider x being object such that
A1:   [x,z] in P*R by XTUPLE_0:def 13;
      consider y such that
A2:   [x,y] in P and
A3:   [y,z] in R by A1,Def6;
      y in rng P by A2,XTUPLE_0:def 13;
      hence thesis by A3,Def11;
    end;
    assume z in R.:(rng P);
    then consider y such that
A4: [y,z] in R and
A5: y in rng P by Def11;
    consider x being object such that
A6: [x,y] in P by A5,XTUPLE_0:def 13;
    [x,z] in P*R by A4,A6,Def6;
    hence thesis by XTUPLE_0:def 13;
  end;
  hence thesis by TARSKI:2;
end;
