let x, y, z be complex number ; :: thesis: ( z = x + y implies Re z = (Re x) + (Re y) )
assume A1: z = x + y ; :: thesis: Re z = (Re x) + (Re 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),(+ x2,y2)*] by XCMPLX_0:def 4;
A5: ( Re x = x1 & Re y = y1 ) by A2, A3, Lm3;
thus Re z = + x1,y1 by A1, A4, Lm3
.= (Re x) + (Re y) by A5, Lm8 ; :: thesis: verum