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

theorem
  z1 <> 0.F_Complex & z2 <> 0.F_Complex implies z1" - z2" = (z2 - z1) *
  (z1 * z2)"
proof
  reconsider z19=z1,z29=z2 as Element of COMPLEX by Def1;
A1: z2 - z1 = z29 - z19 by Th3;
  assume
A2: z1 <> 0.F_Complex;
  then
A3: z1" = z19" by Th5;
  assume
A4: z2 <> 0.F_Complex;
  then z1 * z2 <> 0.F_Complex by A2,VECTSP_1:12;
  then
A5: (z1 * z2)" = (z19 * z29)" by Th5;
  z2" = z29" by A4,Th5;
  hence z1" - z2" = z19" - z29" by A3,Th3
    .= (z2 - z1) * (z1 * z2)" by A2,A4,A1,A5,Th7,XCMPLX_1:212;
end;
