let t be Real; :: thesis: for F, G being real-valued Function st F is t -periodic & G is t -periodic holds
F (#) G is t -periodic
let F, G be real-valued Function; :: thesis: ( F is t -periodic & G is t -periodic implies F (#) G is t -periodic )
assume that
A1: F is t -periodic and
A2: G is t -periodic ; :: thesis: F (#) G is t -periodic
A3: ( t <> 0 & ( for x being Real st x in dom F holds
( x + t in dom F & x - t in dom F & F . x = F . (x + t) ) ) ) by A1, Th1;
for x being Real st x in dom (F (#) G) holds
( x + t in dom (F (#) G) & x - t in dom (F (#) G) & (F (#) G) . x = (F (#) G) . (x + t) )

proof

let x be Real; :: thesis: ( x in dom (F (#) G) implies ( x + t in dom (F (#) G) & x - t in dom (F (#) G) & (F (#) G) . x = (F (#) G) . (x + t) ) )
assume A4: x in dom (F (#) G) ; :: thesis: ( x + t in dom (F (#) G) & x - t in dom (F (#) G) & (F (#) G) . x = (F (#) G) . (x + t) )
then A5: x in (dom F) /\ (dom G) by VALUED_1:def 4;
A6: ( (dom F) /\ (dom G) c= dom F & (dom F) /\ (dom G) c= dom G ) by XBOOLE_1:17;
then A7: ( x + t in dom F & x - t in dom F ) by A1, Th1, A5;
( x + t in dom G & x - t in dom G ) by A2, Th1, A5, A6;
then A8: ( x + t in (dom F) /\ (dom G) & x - t in (dom F) /\ (dom G) ) by A7, XBOOLE_0:def 4;
then A9: ( x + t in dom (F (#) G) & x - t in dom (F (#) G) ) by VALUED_1:def 4;
(F (#) G) . x = (F . x) * (G . x) by A4, VALUED_1:def 4
.= (F . (x + t)) * (G . x) by A1, A5, A6
.= (F . (x + t)) * (G . (x + t)) by A2, A5, A6
.= (F (#) G) . (x + t) by A9, VALUED_1:def 4 ;
hence ( x + t in dom (F (#) G) & x - t in dom (F (#) G) & (F (#) G) . x = (F (#) G) . (x + t) ) by A8, VALUED_1:def 4; :: thesis: verum

end;

hence F (#) G is t -periodic by A3, Th1; :: thesis: verum