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

theorem
  z <> 0.F_Complex implies z1 / z - z2 / z = (z1 - z2) / z
proof
  reconsider z9=z,z19=z1,z29=z2 as Element of COMPLEX by Def1;
A1: z19 - z29 = z1 - z2 by Th3;
  assume
A2: z <> 0.F_Complex;
  then z19 / z9 = z1 / z & z29 / z9 = z2 / z by Th6;
  hence z1 / z - z2 / z = z19 / z9 - z29 / z9 by Th3
    .= (z19 - z29) / z9 by XCMPLX_1:120
    .= (z1 - z2) / z by A2,A1,Th6;
end;
