let A be non empty set ; :: thesis: for f, g being Element of PFuncs (A,REAL) holds (addpfunc A) . (f,g) = (addpfunc A) . (g,f)
let f, g be Element of PFuncs (A,REAL); :: thesis: (addpfunc A) . (f,g) = (addpfunc A) . (g,f)
addpfunc A is commutative by RFUNCT_3:16;
hence (addpfunc A) . (f,g) = (addpfunc A) . (g,f) by BINOP_1:def 2; :: thesis: verum