let x, y, z be Complex; ( z = x + y implies Re z = (Re x) + (Re y) )
assume A1:
z = x + y
; Re z = (Re x) + (Re y)
consider x1, x2, y1, y2 being Element of REAL such that
A2:
( x = [*x1,x2*] & y = [*y1,y2*] )
and
A3:
x + y = [*(+ (x1,y1)),(+ (x2,y2))*]
by XCMPLX_0:def 4;
A4:
( Re x = x1 & Re y = y1 )
by A2, Lm2;
thus Re z =
+ (x1,y1)
by A1, A3, Lm2
.=
(Re x) + (Re y)
by A4, Lm8
; verum