let t be Real; :: 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
then A2: ( 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 Th1;
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) )

proof

let x be Real; :: thesis: ( x in dom |.F.| implies ( x + t in dom |.F.| & x - t in dom |.F.| & |.F.| . x = |.F.| . (x + t) ) )
assume A3: x in dom |.F.| ; :: thesis: ( x + t in dom |.F.| & x - t in dom |.F.| & |.F.| . x = |.F.| . (x + t) )
then A4: x in dom F by VALUED_1:def 11;
then A5: ( x + t in dom F & x - t in dom F ) by A1, Th1;
then A6: ( x + t in dom |.F.| & x - t in dom |.F.| ) by VALUED_1:def 11;
|.F.| . x = |.(F . x).| by A3, VALUED_1:def 11
.= |.(F . (x + t)).| by A1, A4
.= |.F.| . (x + t) by A6, VALUED_1:def 11 ;
hence ( x + t in dom |.F.| & x - t in dom |.F.| & |.F.| . x = |.F.| . (x + t) ) by A5, VALUED_1:def 11; :: thesis: verum

end;

hence |.F.| is t -periodic by A2, Th1; :: thesis: verum