 reserve X, Y for set, A for Ordinal;
 reserve z,z1,z2 for Complex;
 reserve r,r1,r2 for Real;
 reserve q,q1,q2 for Rational;
 reserve i,i1,i2 for Integer;
 reserve n,n1,n2 for Nat;

theorem
  multRel(REAL,r) = the set of all [r1,r1*r] where r1
proof
  set S = the set of all [r1,r1*r] where r1;
  now
    let o be object;
    hereby
      assume A1: o in multRel(REAL,r);
      then consider x,y being object such that
        A2: o = [x,y] by RELAT_1:def 1;
      reconsider x,y as set by TARSKI:1;
      [x,y] in multRel(REAL,r) by A1, A2;
      then x in REAL & y in REAL by MMLQUER2:4;
      then reconsider x, y as Real;
      y = r * x by A1, A2, Th42;
      hence o in S by A2;
    end;
    assume o in S;
    then consider r1 such that
      A3: o = [r1,r1*r];
    r1 in REAL & r1*r in REAL by XREAL_0:def 1;
    hence o in multRel(REAL,r) by A3, Th42;
  end;
  hence thesis by TARSKI:2;
end;
