let X be non empty set ; :: thesis: for Y being ComplexLinearSpace ex MULT being Function of [:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)) st
for c being Complex
for f being Element of Funcs (X, the carrier of Y)
for s being Element of X holds (MULT . [c,f]) . s = c * (f . s)
let Y be ComplexLinearSpace; :: thesis: ex MULT being Function of [:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)) st
for c being Complex
for f being Element of Funcs (X, the carrier of Y)
for s being Element of X holds (MULT . [c,f]) . s = c * (f . s)
deffunc H₁( Complex, Element of Funcs (X, the carrier of Y)) -> Element of Funcs (X, the carrier of Y) = the Mult of Y [;] ($1,$2);
consider F being Function of [:COMPLEX,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)) such that
A1: for c being Complex
for f being Element of Funcs (X, the carrier of Y) holds F . (c,f) = H₁(c,f) from BINOP_1:sch 4();
take F ; :: thesis: for c being Complex
for f being Element of Funcs (X, the carrier of Y)
for s being Element of X holds (F . [c,f]) . s = c * (f . s)
let c be Complex; :: thesis: for f being Element of Funcs (X, the carrier of Y)
for s being Element of X holds (F . [c,f]) . s = c * (f . s)
let f be Element of Funcs (X, the carrier of Y); :: thesis: for s being Element of X holds (F . [c,f]) . s = c * (f . s)
let x be Element of X; :: thesis: (F . [c,f]) . x = c * (f . x)
A2: dom (F . [c,f]) = X by FUNCT_2:169;
F . (c,f) = the Mult of Y [;] (c,f) by A1;
hence (F . [c,f]) . x = the Mult of Y . (c,(f . x)) by A2, FUNCOP_1:42
.= c * (f . x) by CLVECT_1:def 1 ;
:: thesis: verum