reserve z,z1,z2,z3,z4 for Element of F_Complex;

theorem
  z1 <> z2 iff 0 < |.z1 - z2.|
proof
  reconsider z19=z1,z29=z2 as Element of COMPLEX by Def1;
  thus z1 <> z2 implies 0 < |.z1 - z2.|
  proof
    assume
A1: z1 <> z2;
    z1 - z2 = z19 - z29 by Th3;
    hence thesis by A1,COMPLEX1:62;
  end;
  assume
A2: 0 < |.z1 - z2.|;
  z1 - z2 = z19 - z29 by Th3;
  hence thesis by A2;
end;
