let A be non empty set ; :: thesis: for f, g being Element of Funcs (A,COMPLEX) holds (ComplexFuncAdd A) . (f,g) = (ComplexFuncAdd A) . (g,f)

let f, g be Element of Funcs (A,COMPLEX); :: thesis: (ComplexFuncAdd A) . (f,g) = (ComplexFuncAdd A) . (g,f)

let f, g be Element of Funcs (A,COMPLEX); :: thesis: (ComplexFuncAdd A) . (f,g) = (ComplexFuncAdd A) . (g,f)

now :: thesis: for x being Element of A holds ((ComplexFuncAdd A) . (f,g)) . x = ((ComplexFuncAdd A) . (g,f)) . x

hence
(ComplexFuncAdd A) . (f,g) = (ComplexFuncAdd A) . (g,f)
by FUNCT_2:63; :: thesis: verumlet x be Element of A; :: thesis: ((ComplexFuncAdd A) . (f,g)) . x = ((ComplexFuncAdd A) . (g,f)) . x

thus ((ComplexFuncAdd A) . (f,g)) . x = (g . x) + (f . x) by Th1

.= ((ComplexFuncAdd A) . (g,f)) . x by Th1 ; :: thesis: verum

end;thus ((ComplexFuncAdd A) . (f,g)) . x = (g . x) + (f . x) by Th1

.= ((ComplexFuncAdd A) . (g,f)) . x by Th1 ; :: thesis: verum