let X be non empty set ; :: thesis: for Y being RealLinearSpace ex MULT being Function of [:REAL ,(Funcs X,the carrier of Y):],(Funcs X,the carrier of Y) st
for r being Real
for f being Element of Funcs X,the carrier of Y
for s being Element of X holds (MULT . [r,f]) . s = r * (f . s)
let Y be RealLinearSpace; :: thesis: ex MULT being Function of [:REAL ,(Funcs X,the carrier of Y):],(Funcs X,the carrier of Y) st
for r being Real
for f being Element of Funcs X,the carrier of Y
for s being Element of X holds (MULT . [r,f]) . s = r * (f . s)
deffunc H₁( Real, 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 [:REAL ,(Funcs X,the carrier of Y):],(Funcs X,the carrier of Y) such that
A1: for a being Real
for f being Element of Funcs X,the carrier of Y holds F . a,f = H₁(a,f) from BINOP_1:sch 4();
take F ; :: thesis: for r being Real
for f being Element of Funcs X,the carrier of Y
for s being Element of X holds (F . [r,f]) . s = r * (f . s)
let a be Real; :: thesis: for f being Element of Funcs X,the carrier of Y
for s being Element of X holds (F . [a,f]) . s = a * (f . s)
let f be Element of Funcs X,the carrier of Y; :: thesis: for s being Element of X holds (F . [a,f]) . s = a * (f . s)
let x be Element of X; :: thesis: (F . [a,f]) . x = a * (f . x)
A2: dom (F . [a,f]) = X by FUNCT_2:169;
F . a,f = the Mult of Y [;] a,f by A1;
hence (F . [a,f]) . x = a * (f . x) by A2, FUNCOP_1:42; :: thesis: verum