 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
  for X being real-membered set
  holds multRel(X,r) = ((curry multreal).r) |_2 X
proof
  let X be real-membered set;
  set g = (curry multreal).r;
  now
    let x,y be object;
    reconsider a=x,b=y as set by TARSKI:1;
    hereby
      assume A1: [x,y] in multRel(X,r);
      then [a,b] in multRel(X,r);
      then a in X & b in X by MMLQUER2:4;
      then reconsider a,b as Real;
      [a,b] in multRel(X,r) by A1;
      then A2: a in X & b in X & b = r * a by Th42;
      r in REAL & a in REAL by XREAL_0:def 1;
      then [r,a] in [: REAL, REAL :] by ZFMISC_1:87;
      then A3: [r,a] in dom multreal by FUNCT_2:def 1;
      A4: b = multreal.(r,a) by A2, BINOP_2:def 11
        .= g.a by A3, FUNCT_5:20;
      a in dom g by A3, FUNCT_5:20;
      hence [x,y] in g |_2 X by A2, A4, FUNCT_1:1, MMLQUER2:4;
    end;
    assume [x,y] in g |_2 X;
    then [a,b] in g |_2 X;
    then A5: a in X & b in X & [a,b] in g by MMLQUER2:4;
    then reconsider a,b as Real;
    r in REAL & a in REAL by XREAL_0:def 1;
    then [r,a] in [: REAL, REAL :] by ZFMISC_1:87;
    then A6: [r,a] in dom multreal by FUNCT_2:def 1;
    a in dom g & g.a = b by A5, FUNCT_1:1;
    then b = multreal.(r,a) by A6, FUNCT_5:20
     .= r * a by BINOP_2:def 11;
    hence [x,y] in multRel(X,r) by A5, Th42;
  end;
  hence thesis by RELAT_1:def 2;
end;
