 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
  addRel(COMPLEX,z1) * addRel(COMPLEX,z2) = addRel(COMPLEX,z1+z2)
proof
  A1: addRel(COMPLEX,z1) * addRel(COMPLEX,z2) c= addRel(COMPLEX,z1+z2) by Th17;
  now
    let x,y be object;
    reconsider a=x,b=y as set by TARSKI:1;
    assume A2: [x,y] in addRel(COMPLEX,z1+z2);
    then [a,b] in addRel(COMPLEX,z1+z2);
    then A3: a in COMPLEX & b in COMPLEX by MMLQUER2:4;
    then reconsider a,b as Complex;
    A4: b = z1 + z2 + a by A2, Th11;
    set c = z1 + a;
    c in COMPLEX & b = z2 + c by A4, XCMPLX_0:def 2;
    then [a,c] in addRel(COMPLEX,z1) & [c,b] in addRel(COMPLEX,z2) by A3, Th11;
    hence [x,y] in addRel(COMPLEX,z1) * addRel(COMPLEX,z2) by RELAT_1:def 8;
  end;
  then addRel(COMPLEX,z1+z2) c= addRel(COMPLEX,z1) * addRel(COMPLEX,z2)
    by RELAT_1:def 3;
  hence thesis by A1, XBOOLE_0:def 10;
end;
