let x, y, z be complex number ; :: thesis: ( z = x * y implies Re z = ((Re x) * (Re y)) - ((Im x) * (Im y)) )
assume A1: z = x * y ; :: thesis: Re z = ((Re x) * (Re y)) - ((Im x) * (Im y))
consider x1, x2, y1, y2 being Element of REAL such that
A2: x = [*x1,x2*] and
A3: y = [*y1,y2*] and
A4: x * y = [*(+ (* x1,y1),(opp (* x2,y2))),(+ (* x1,y2),(* x2,y1))*] by XCMPLX_0:def 5;
A5: ( Re x = x1 & Re y = y1 ) by A2, A3, Lm3;
A6: ( Im x = x2 & Im y = y2 ) by A2, A3, Lm3;
thus Re z = + (* x1,y1),(opp (* x2,y2)) by A1, A4, Lm3
.= (* x1,y1) + (opp (* x2,y2)) by Lm8
.= (x1 * y1) + (opp (* x2,y2)) by Lm10
.= (x1 * y1) + (- (* x2,y2)) by Lm9
.= (x1 * y1) - (* x2,y2)
.= ((Re x) * (Re y)) - ((Im x) * (Im y)) by A5, A6, Lm10 ; :: thesis: verum