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

theorem
  z2 <> 0.F_Complex & (z1 * z2 = 1.F_Complex or z2 * z1 = 1.F_Complex)
  implies z1 = z2"
proof
  reconsider z19=z1,z29=z2 as Element of COMPLEX by Def1;
  assume
A1: z2 <> 0.F_Complex;
  assume
A2: z1 * z2 = 1.F_Complex or z2 * z1 = 1.F_Complex;
  per cases by A2;
  suppose
A3: z1 * z2 = 1.F_Complex;
A4: z2" = z29" by A1,Th5;
    z19 * z29 = 1r by A3,Def1;
    hence thesis by A1,A4,Th7,XCMPLX_0:def 7;
  end;
  suppose
A5: z2 * z1 = 1.F_Complex;
A6: z2" = z29" by A1,Th5;
    z29 * z19 = 1r by A5,Def1;
    hence thesis by A1,A6,Th7,XCMPLX_0:def 7;
  end;
end;
