let X be non empty set ; :: thesis: for Y being RealLinearSpace
for f being Element of Funcs (X, the carrier of Y)
for a, b being Real holds (FuncAdd (X,Y)) . (((FuncExtMult (X,Y)) . [a,f]),((FuncExtMult (X,Y)) . [b,f])) = (FuncExtMult (X,Y)) . [(a + b),f]
let Y be RealLinearSpace; :: thesis: for f being Element of Funcs (X, the carrier of Y)
for a, b being Real holds (FuncAdd (X,Y)) . (((FuncExtMult (X,Y)) . [a,f]),((FuncExtMult (X,Y)) . [b,f])) = (FuncExtMult (X,Y)) . [(a + b),f]
let f be Element of Funcs (X, the carrier of Y); :: thesis: for a, b being Real holds (FuncAdd (X,Y)) . (((FuncExtMult (X,Y)) . [a,f]),((FuncExtMult (X,Y)) . [b,f])) = (FuncExtMult (X,Y)) . [(a + b),f]
let a, b be Real; :: thesis: (FuncAdd (X,Y)) . (((FuncExtMult (X,Y)) . [a,f]),((FuncExtMult (X,Y)) . [b,f])) = (FuncExtMult (X,Y)) . [(a + b),f]
reconsider a = a, b = b as Element of REAL by XREAL_0:def 1;

now :: thesis: for x being Element of X holds ((FuncAdd (X,Y)) . (((FuncExtMult (X,Y)) . [a,f]),((FuncExtMult (X,Y)) . [b,f]))) . x = ((FuncExtMult (X,Y)) . [(a + b),f]) . x

let x be Element of X; :: thesis: ((FuncAdd (X,Y)) . (((FuncExtMult (X,Y)) . [a,f]),((FuncExtMult (X,Y)) . [b,f]))) . x = ((FuncExtMult (X,Y)) . [(a + b),f]) . x
thus ((FuncAdd (X,Y)) . (((FuncExtMult (X,Y)) . [a,f]),((FuncExtMult (X,Y)) . [b,f]))) . x = (((FuncExtMult (X,Y)) . [a,f]) . x) + (((FuncExtMult (X,Y)) . [b,f]) . x) by Th1
.= (a * (f . x)) + (((FuncExtMult (X,Y)) . [b,f]) . x) by Th2
.= (a * (f . x)) + (b * (f . x)) by Th2
.= (a + b) * (f . x) by RLVECT_1:def 6
.= ((FuncExtMult (X,Y)) . [(a + b),f]) . x by Th2 ; :: thesis: verum

end;

hence (FuncAdd (X,Y)) . (((FuncExtMult (X,Y)) . [a,f]),((FuncExtMult (X,Y)) . [b,f])) = (FuncExtMult (X,Y)) . [(a + b),f] by FUNCT_2:63; :: thesis: verum