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 Th11:
  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 7; 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 7;
        F".x=(F.x)" by A3,VALUED_1:def 7
                 .=(F.(x+t))" by A1,A4
                 .=F".(x+t) by A6,VALUED_1:def 7;
          hence thesis by A5,VALUED_1:def 7;
      end;
 hence thesis by A2,Th1;
end;
