let x1, y1 be FinSequence of COMPLEX ; for x2, y2 being FinSequence of REAL st x1 = x2 & y1 = y2 & len x1 = len y2 holds
multcomplex .: (x1,y1) = multreal .: (x2,y2)
let x2, y2 be FinSequence of REAL ; ( x1 = x2 & y1 = y2 & len x1 = len y2 implies multcomplex .: (x1,y1) = multreal .: (x2,y2) )
assume that
A1:
x1 = x2
and
A2:
y1 = y2
and
A3:
len x1 = len y2
; multcomplex .: (x1,y1) = multreal .: (x2,y2)
for i being Element of NAT st i in dom x1 holds
multcomplex . ((x1 . i),(y1 . i)) = multreal . ((x2 . i),(y2 . i))
hence
multcomplex .: (x1,y1) = multreal .: (x2,y2)
by A1, A2, A3, COMPLSP2:45; verum