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 Th7:
  F is t-periodic implies r (#) 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 (r (#) F) holds (x+t in dom (r (#) F)
    & x-t in dom (r (#) F)) & (r (#) F).x=(r (#) F).(x+t)
      proof
        let x;
        assume
A3:     x in dom (r (#) F); then
A4:     x in dom F by VALUED_1:def 5; then
A5:    x+t in dom F & x-t in dom F by A1,Th1; then
A6:     x+t in dom (r (#) F) & x-t in dom (r (#) F) by VALUED_1:def 5;
        (r (#) F).x=r * F.x by A3,VALUED_1:def 5
                 .=r * F.(x+t) by A1,A4
                 .=(r (#) F).(x+t) by A6,VALUED_1:def 5;
          hence thesis by A5,VALUED_1:def 5;
      end;
 hence thesis by A2,Th1;
end;
