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 Th8:
  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 2; 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 2;
        (r+F).x=r + F.x by A3,VALUED_1:def 2
                 .=r + F.(x+t) by A1,A4
                 .=(r+F).(x+t) by A6,VALUED_1:def 2;
          hence thesis by A5,VALUED_1:def 2;
      end;
   hence thesis by A2,Th1;
end;
