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)

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

let 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;

hence (ComplexFuncAdd A) . (f,g) = (ComplexFuncAdd A) . (g,f) by FUNCT_2:63; :: thesis: verum