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)

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

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