let t be real number ; :: thesis: for F being real-valued Function st F is t -periodic holds
F " is t -periodic
let F be real-valued Function; :: thesis: ( F is t -periodic implies F " is t -periodic )
assume A1: F is t -periodic ; :: thesis: F " is t -periodic
A3: ( t <> 0 & ( for x being real number 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 number st x in dom (F " ) holds
( x + t in dom (F " ) & x - t in dom (F " ) & (F " ) . x = (F " ) . (x + t) )

proof

let x be real number ; :: thesis: ( x in dom (F " ) implies ( x + t in dom (F " ) & x - t in dom (F " ) & (F " ) . x = (F " ) . (x + t) ) )
assume B1: x in dom (F " ) ; :: thesis: ( x + t in dom (F " ) & x - t in dom (F " ) & (F " ) . x = (F " ) . (x + t) )
then A5: x in dom F by VALUED_1:def 7;
then A6: ( x + t in dom F & x - t in dom F ) by A1, Th1;
then A8: ( x + t in dom (F " ) & x - t in dom (F " ) ) by VALUED_1:def 7;
(F " ) . x = (F . x) " by B1, VALUED_1:def 7
.= (F . (x + t)) " by A1, Th1, A5
.= (F " ) . (x + t) by A8, VALUED_1:def 7 ;
hence ( x + t in dom (F " ) & x - t in dom (F " ) & (F " ) . x = (F " ) . (x + t) ) by A6, VALUED_1:def 7; :: thesis: verum

end;

hence F " is t -periodic by A3, Th1; :: thesis: verum