let z1, z2 be Element of F_Complex ; :: thesis: ( z1 <> 0. F_Complex & z2 <> 0. F_Complex implies (z1 " ) + (z2 " ) = (z1 + z2) * ((z1 * z2) " ) )
reconsider z19 = z1, z29 = z2 as Element of COMPLEX by Def1;
assume A1: z1 <> 0. F_Complex ; :: thesis: ( not z2 <> 0. F_Complex or (z1 " ) + (z2 " ) = (z1 + z2) * ((z1 * z2) " ) )
then A2: z1 " = z19 " by Th7;
assume A3: z2 <> 0. F_Complex ; :: thesis: (z1 " ) + (z2 " ) = (z1 + z2) * ((z1 * z2) " )
then z1 * z2 <> 0. F_Complex by A1, VECTSP_1:44;
then A4: (z1 * z2) " = (z19 * z29) " by Th7;
z2 " = z29 " by A3, Th7;
hence (z1 " ) + (z2 " ) = (z1 + z2) * ((z1 * z2) " ) by A1, A2, A3, A4, Th9, XCMPLX_1:213; :: thesis: verum