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

theorem
  |.z1 - z2.| = 0 iff z1 = z2
proof
  reconsider z19=z1,z29=z2 as Element of COMPLEX by Def1;
  thus |.z1 - z2.| = 0 implies z1 = z2
  proof
    assume
A1: |.z1 - z2.| = 0;
    z1 - z2 = z19 - z29 by Th3;
    hence thesis by A1,COMPLEX1:61;
  end;
  assume
A2: z1 = z2;
  z19 - z29 = z1 - z2 by Th3;
  hence thesis by A2;
end;
