reserve x,t,t1,t2,r,a,b for Real;
reserve F,G for real-valued Function;
reserve k for Nat;
reserve i for non zero Integer;

theorem Th10:
  F is t-periodic implies |. F .| is t-periodic
proof
    assume
A1: F is t-periodic;
then A2: t<>0 & for x 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 st x in dom |. F .| holds (x+t in dom |. F .| & x-t in dom |. F .|) &
   |. F .|.x=|. F .|.(x+t)
    proof
      let x;
      assume
A3:   x in dom |. F .|; 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 thesis by A5,VALUED_1:def 11;
    end;
    hence thesis by A2,Th1;
end;
