let z1, z2 be Element of F_Complex; :: thesis: ( z1 <> 0. F_Complex & z2 <> 0. F_Complex implies (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 ") )
then A2: z1 " = z19 " by Th5;
assume A3: z2 <> 0. F_Complex ; :: thesis: (z1 * z2) " = (z1 ") * (z2 ")
then A4: z2 " = z29 " by Th5;
z1 * z2 <> 0. F_Complex by A1, A3, VECTSP_1:12;
hence (z1 * z2) " = (z19 * z29) " by Th5
.= (z1 ") * (z2 ") by A2, A4, XCMPLX_1:204 ;
:: thesis: verum