let X be non empty set ; :: thesis: for Y being ComplexLinearSpace
for f, g being Element of Funcs (X, the carrier of Y) holds (FuncAdd (X,Y)) . (f,g) = (FuncAdd (X,Y)) . (g,f)
let Y be ComplexLinearSpace; :: thesis: for f, g being Element of Funcs (X, the carrier of Y) holds (FuncAdd (X,Y)) . (f,g) = (FuncAdd (X,Y)) . (g,f)
let f, g be Element of Funcs (X, the carrier of Y); :: thesis: (FuncAdd (X,Y)) . (f,g) = (FuncAdd (X,Y)) . (g,f)

now :: thesis: for x being Element of X holds ((FuncAdd (X,Y)) . (f,g)) . x = ((FuncAdd (X,Y)) . (g,f)) . x

let x be Element of X; :: thesis: ((FuncAdd (X,Y)) . (f,g)) . x = ((FuncAdd (X,Y)) . (g,f)) . x
thus ((FuncAdd (X,Y)) . (f,g)) . x = (g . x) + (f . x) by LOPBAN_1:1
.= ((FuncAdd (X,Y)) . (g,f)) . x by LOPBAN_1:1 ; :: thesis: verum

end;

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