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

theorem
  z1 <> 0.F_Complex & z2 <> 0.F_Complex implies z1" + z2" = (z1 + z2) *
  (z1 * z2)"
proof
  reconsider z19=z1,z29=z2 as Element of COMPLEX by Def1;
  assume
A1: z1 <> 0.F_Complex;
  then
A2: z1" = z19" by Th5;
  assume
A3: z2 <> 0.F_Complex;
  then z1 * z2 <> 0.F_Complex by A1,VECTSP_1:12;
  then
A4: (z1 * z2)" = (z19 * z29)" by Th5;
  z2" = z29" by A3,Th5;
  hence thesis by A1,A2,A3,A4,Th7,XCMPLX_1:211;
end;
