 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 Th11:
  for X being complex-membered set
  holds [z1,z2] in addRel(X,z) iff z1 in X & z2 in X & z2 = z + z1
proof
  let X be complex-membered set;
  reconsider g = (curry addcomplex).z as Function;
  z in COMPLEX & z1 in COMPLEX by XCMPLX_0:def 2;
  then [z,z1] in [: COMPLEX, COMPLEX :] by ZFMISC_1:87;
  then A1: [z,z1] in dom addcomplex by FUNCT_2:def 1;
  hereby
    assume [z1,z2] in addRel(X,z);
    then [z1,z2] in g |_2 X & z1 is set & z2 is set by TARSKI:1;
    then A2: z1 in X & z2 in X & [z1,z2] in g by MMLQUER2:4;
    then z1 in dom g & g.z1 = z2 by FUNCT_1:1;
    then z2 = addcomplex.(z,z1) by A1, FUNCT_5:20;
    hence z1 in X & z2 in X & z2 = z + z1 by A2, BINOP_2:def 3;
  end;
  assume A3: z1 in X & z2 in X & z2 = z + z1;
  then A4: z2 = addcomplex.(z,z1) by BINOP_2:def 3
    .= g.z1 by A1, FUNCT_5:20;
  z1 in dom g by A1, FUNCT_5:20;
  hence thesis by A3, A4, FUNCT_1:1, MMLQUER2:4;
end;
