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

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_1:3
.= ((FuncAdd X,Y) . g,f) . x by LOPBAN_1:3 ; :: thesis: verum

end;

hence (FuncAdd X,Y) . f,g = (FuncAdd X,Y) . g,f by FUNCT_2:113; :: thesis: verum