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 ;
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 1;
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 ;
then A9: ( x + t in dom (F + G) & x - t in dom (F + G) ) by VALUED_1:def 1;
(F + G) . x = (F . x) + (G . x) by
.= (F . (x + t)) + (G . x) by A1, A5, A6
.= (F . (x + t)) + (G . (x + t)) by A2, A5, A6
.= (F + G) . (x + t) by ;
hence ( x + t in dom (F + G) & x - t in dom (F + G) & (F + G) . x = (F + G) . (x + t) ) by ; :: thesis: verum
end;
hence F + G is t -periodic by ; :: thesis: verum