let x1, y1 be FinSequence of COMPLEX ; :: thesis: for x2, y2 being FinSequence of REAL st x1 = x2 & y1 = y2 & len x1 = len y2 holds
diffcomplex .: x1,y1 = diffreal .: x2,y2
let x2, y2 be FinSequence of REAL ; :: thesis: ( x1 = x2 & y1 = y2 & len x1 = len y2 implies diffcomplex .: x1,y1 = diffreal .: x2,y2 )
assume that
A1: ( x1 = x2 & y1 = y2 ) and
A2: len x1 = len y2 ; :: thesis: diffcomplex .: x1,y1 = diffreal .: x2,y2
for i being Element of NAT st i in dom x1 holds
diffcomplex . (x1 . i),(y1 . i) = diffreal . (x2 . i),(y2 . i)

proof

let i be Element of NAT ; :: thesis: ( i in dom x1 implies diffcomplex . (x1 . i),(y1 . i) = diffreal . (x2 . i),(y2 . i) )
(x1 . i) - (y1 . i) = diffcomplex . (x1 . i),(y1 . i) by BINOP_2:def 4;
hence ( i in dom x1 implies diffcomplex . (x1 . i),(y1 . i) = diffreal . (x2 . i),(y2 . i) ) by A1, BINOP_2:def 10; :: thesis: verum

end;

hence diffcomplex .: x1,y1 = diffreal .: x2,y2 by A1, A2, Th45; :: thesis: verum