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

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 LOPBAN_1:1
.= (0. Y) + (f . x) by Th1
.= f . x by RLVECT_1:def 4 ; :: thesis: verum

end;

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