 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 natural-membered set
  holds multRel(X,n) = ((curry multnat).n) |_2 X
proof
  let X be natural-membered set;
  set g = (curry multnat).n;
  now
    let x,y be object;
    reconsider a=x,b=y as set by TARSKI:1;
    hereby
      assume A1: [x,y] in multRel(X,n);
      then [a,b] in multRel(X,n);
      then a in X & b in X by MMLQUER2:4;
      then reconsider a,b as Integer;
      [a,b] in multRel(X,n) by A1;
      then A2: a in X & b in X & b = n * a by Th42;
      then n in NAT & a in NAT by ORDINAL1:def 12;
      then [n,a] in [: NAT, NAT :] by ZFMISC_1:87;
      then A3: [n,a] in dom multnat by FUNCT_2:def 1;
      A4: b = multnat.(n,a) by A2, BINOP_2:def 24
        .= 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 Integer;
    A6: n in NAT & a in NAT by A5, ORDINAL1:def 12;
    then [n,a] in [: NAT, NAT :] by ZFMISC_1:87;
    then A7: [n,a] in dom multnat by FUNCT_2:def 1;
    a in dom g & g.a = b by A5, FUNCT_1:1;
    then b = multnat.(n,a) by A7, FUNCT_5:20
     .= n * a by A6, BINOP_2:def 24;
    hence [x,y] in multRel(X,n) by A5, Th42;
  end;
  hence thesis by RELAT_1:def 2;
end;
