let it1, it2 be Function of [:REAL,(Funcs (X, the carrier of Y)):],(Funcs (X, the carrier of Y)); :: thesis: ( ( for a being Real
for f being Element of Funcs (X, the carrier of Y)
for x being Element of X holds (it1 . [a,f]) . x = a * (f . x) ) & ( for a being Real
for f being Element of Funcs (X, the carrier of Y)
for x being Element of X holds (it2 . [a,f]) . x = a * (f . x) ) implies it1 = it2 )
assume that
A3: for a being Real
for f being Element of Funcs (X, the carrier of Y)
for x being Element of X holds (it1 . [a,f]) . x = a * (f . x) and
A4: for a being Real
for f being Element of Funcs (X, the carrier of Y)
for x being Element of X holds (it2 . [a,f]) . x = a * (f . x) ; :: thesis: it1 = it2

now :: thesis: for a being Element of REAL
for f being Element of Funcs (X, the carrier of Y) holds it1 . (a,f) = it2 . (a,f)

let a be Element of REAL ; :: thesis: for f being Element of Funcs (X, the carrier of Y) holds it1 . (a,f) = it2 . (a,f)
let f be Element of Funcs (X, the carrier of Y); :: thesis: it1 . (a,f) = it2 . (a,f)

now :: thesis: for x being Element of X holds (it1 . [a,f]) . x = (it2 . [a,f]) . x

let x be Element of X; :: thesis: (it1 . [a,f]) . x = (it2 . [a,f]) . x
thus (it1 . [a,f]) . x = a * (f . x) by A3
.= (it2 . [a,f]) . x by A4 ; :: thesis: verum

end;

hence it1 . (a,f) = it2 . (a,f) by FUNCT_2:63; :: thesis: verum

end;

hence it1 = it2 ; :: thesis: verum