let C be Function of ,COMPLEX; :: thesis: for G being Function of ,REAL
for x1, y1 being FinSequence of COMPLEX
for x2, y2 being FinSequence of REAL st x1 = x2 & y1 = y2 & len x1 = len y2 & ( for i being Element of NAT st i in dom x1 holds
C . ((x1 . i),(y1 . i)) = G . ((x2 . i),(y2 . i)) ) holds
C .: (x1,y1) = G .: (x2,y2)

let G be Function of ,REAL; :: thesis: for x1, y1 being FinSequence of COMPLEX
for x2, y2 being FinSequence of REAL st x1 = x2 & y1 = y2 & len x1 = len y2 & ( for i being Element of NAT st i in dom x1 holds
C . ((x1 . i),(y1 . i)) = G . ((x2 . i),(y2 . i)) ) holds
C .: (x1,y1) = G .: (x2,y2)

let x1, y1 be FinSequence of COMPLEX ; :: thesis: for x2, y2 being FinSequence of REAL st x1 = x2 & y1 = y2 & len x1 = len y2 & ( for i being Element of NAT st i in dom x1 holds
C . ((x1 . i),(y1 . i)) = G . ((x2 . i),(y2 . i)) ) holds
C .: (x1,y1) = G .: (x2,y2)

let x2, y2 be FinSequence of REAL ; :: thesis: ( x1 = x2 & y1 = y2 & len x1 = len y2 & ( for i being Element of NAT st i in dom x1 holds
C . ((x1 . i),(y1 . i)) = G . ((x2 . i),(y2 . i)) ) implies C .: (x1,y1) = G .: (x2,y2) )

assume that
A1: x1 = x2 and
A2: y1 = y2 and
A3: len x1 = len y2 and
A4: for i being Element of NAT st i in dom x1 holds
C . ((x1 . i),(y1 . i)) = G . ((x2 . i),(y2 . i)) ; :: thesis: C .: (x1,y1) = G .: (x2,y2)
A5: len (G .: (x2,y2)) = len x1 by ;
now :: thesis: for i being Nat st 1 <= i & i <= len (C .: (x1,y1)) holds
(C .: (x1,y1)) . i = (G .: (x2,y2)) . i
let i be Nat; :: thesis: ( 1 <= i & i <= len (C .: (x1,y1)) implies (C .: (x1,y1)) . i = (G .: (x2,y2)) . i )
assume that
A6: 1 <= i and
A7: i <= len (C .: (x1,y1)) ; :: thesis: (C .: (x1,y1)) . i = (G .: (x2,y2)) . i
A8: i <= len x1 by ;
then A9: i in dom x1 by ;
A10: i in dom (G .: (x2,y2)) by ;
i in dom (C .: (x1,y1)) by ;
hence (C .: (x1,y1)) . i = C . ((x1 . i),(y1 . i)) by FUNCOP_1:22
.= G . ((x2 . i),(y2 . i)) by A4, A9
.= (G .: (x2,y2)) . i by ;
:: thesis: verum
end;
hence C .: (x1,y1) = G .: (x2,y2) by ; :: thesis: verum