let X be non empty set ; :: thesis: for Y being ComplexLinearSpace

for f being Element of Funcs (X, the carrier of Y) holds (FuncAdd (X,Y)) . ((FuncZero (X,Y)),f) = f

let Y be ComplexLinearSpace; :: 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

for f being Element of Funcs (X, the carrier of Y) holds (FuncAdd (X,Y)) . ((FuncZero (X,Y)),f) = f

let Y be ComplexLinearSpace; :: 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

hence
(FuncAdd (X,Y)) . ((FuncZero (X,Y)),f) = f
by FUNCT_2:63; :: thesis: verumlet 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 LOPBAN_1:1

.= (0. Y) + (f . x) by Th1

.= f . x by RLVECT_1:def 4 ; :: thesis: verum

end;thus ((FuncAdd (X,Y)) . ((FuncZero (X,Y)),f)) . x = ((FuncZero (X,Y)) . x) + (f . x) by LOPBAN_1:1

.= (0. Y) + (f . x) by Th1

.= f . x by RLVECT_1:def 4 ; :: thesis: verum