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

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

let y1 be FinSequence of COMPLEX ; :: thesis: for y2 being FinSequence of REAL st len y1 = len y2 & ( for i being Element of NAT st i in dom y1 holds
h . (y1 . i) = g . (y2 . i) ) holds
h * y1 = g * y2

let y2 be FinSequence of REAL ; :: thesis: ( len y1 = len y2 & ( for i being Element of NAT st i in dom y1 holds
h . (y1 . i) = g . (y2 . i) ) implies h * y1 = g * y2 )

assume that
A1: len y1 = len y2 and
A2: for i being Element of NAT st i in dom y1 holds
h . (y1 . i) = g . (y2 . i) ; :: thesis: h * y1 = g * y2
A3: len (g * y2) = len y1 by ;
now :: thesis: for i being Nat st 1 <= i & i <= len (h * y1) holds
(h * y1) . i = (g * y2) . i
let i be Nat; :: thesis: ( 1 <= i & i <= len (h * y1) implies (h * y1) . i = (g * y2) . i )
assume that
A4: 1 <= i and
A5: i <= len (h * y1) ; :: thesis: (h * y1) . i = (g * y2) . i
A6: i <= len y1 by ;
then A7: i in dom y1 by ;
A8: i in dom (g * y2) by ;
i in dom (h * y1) by ;
hence (h * y1) . i = h . (y1 . i) by FUNCT_1:12
.= g . (y2 . i) by A2, A7
.= (g * y2) . i by ;
:: thesis: verum
end;
hence h * y1 = g * y2 by ; :: thesis: verum