let z1, z2, z be Element of F_Complex ; :: thesis: ( z1 <> 0. F_Complex & z2 <> 0. F_Complex implies (z1 " ) * (z / z2) = z / (z1 * z2) )
reconsider z1' = z1, z2' = z2, z' = z as Element of COMPLEX by Def1;
assume A1: z1 <> 0. F_Complex ; :: thesis: ( not z2 <> 0. F_Complex or (z1 " ) * (z / z2) = z / (z1 * z2) )
assume A2: z2 <> 0. F_Complex ; :: thesis: (z1 " ) * (z / z2) = z / (z1 * z2)
then A3: z1 * z2 <> 0. F_Complex by A1, VECTSP_1:44;
A4: z1 " = z1' " by A1, Th7;
z' / z2' = z / z2 by A2, Th8;
hence (z1 " ) * (z / z2) = z' / (z1' * z2') by A4, XCMPLX_1:222
.= z / (z1 * z2) by A3, Th8 ;
:: thesis: verum