let z1, z2 be Complex; :: thesis: ( Im z1 = 0 & Im z2 = 0 & Re z2 <> 0 implies ( Re (z1 / z2) = (Re z1) / (Re z2) & Im (z1 / z2) = 0 ) )
assume that
A1: Im z1 = 0 and
A2: ( Im z2 = 0 & Re z2 <> 0 ) ; :: thesis: ( Re (z1 / z2) = (Re z1) / (Re z2) & Im (z1 / z2) = 0 )
A3: ( z1 / z2 = z1 * (z2 ") & Im (z2 ") = 0 ) by A2, Th22, XCMPLX_0:def 9;
hence Re (z1 / z2) = (Re z1) * (Re (z2 ")) by A1, Th14
.= (Re z1) * ((Re z2) ") by A2, Th22
.= (Re z1) / (Re z2) by XCMPLX_0:def 9 ;
:: thesis: Im (z1 / z2) = 0
thus Im (z1 / z2) = 0 by A1, A3, Th14; :: thesis: verum