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 A1, FINSEQ_2:37;

A4: now

let i be Nat; :: thesis: ( 1 <= i & i <= len (h * y1) implies (h * y1) . i = (g * y2) . i )
assume that
A5: 1 <= i and
A6: i <= len (h * y1) ; :: thesis: (h * y1) . i = (g * y2) . i
A7: i <= len y1 by A6, FINSEQ_2:37;
then A8: i in dom y1 by A5, FINSEQ_3:27;
A9: i in dom (g * y2) by A3, A5, A7, FINSEQ_3:27;
i in dom (h * y1) by A5, A6, FINSEQ_3:27;
hence (h * y1) . i = h . (y1 . i) by FUNCT_1:22
.= g . (y2 . i) by A2, A8
.= (g * y2) . i by A9, FUNCT_1:22 ;
:: thesis: verum

end;

len (h * y1) = len y1 by FINSEQ_2:37;
hence h * y1 = g * y2 by A3, A4, FINSEQ_1:18; :: thesis: verum