let X be non empty set ; :: thesis: for Y being RealLinearSpace
for f being Element of Funcs (X, the carrier of Y) holds (FuncAdd (X,Y)) . ((FuncZero (X,Y)),f) = f
let Y be RealLinearSpace; :: thesis: for f being Element of Funcs (X, the carrier of Y) holds (FuncAdd (X,Y)) . ((FuncZero (X,Y)),f) = f
let f be Element of Funcs (X, the carrier of Y); :: thesis: (FuncAdd (X,Y)) . ((FuncZero (X,Y)),f) = f
now :: thesis: for x being Element of X holds ((FuncAdd (X,Y)) . ((FuncZero (X,Y)),f)) . x = f . x
let x be Element of X; :: thesis: ((FuncAdd (X,Y)) . ((FuncZero (X,Y)),f)) . x = f . x
thus ((FuncAdd (X,Y)) . ((FuncZero (X,Y)),f)) . x = ((FuncZero (X,Y)) . x) + (f . x) by Th1
.= (0. Y) + (f . x) by FUNCOP_1:7
.= f . x ; :: thesis: verum
end;
hence (FuncAdd (X,Y)) . ((FuncZero (X,Y)),f) = f by FUNCT_2:63; :: thesis: verum