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
  (t<>0 & for x st x in dom F holds (x+t in dom F & x-t in dom F &
  F.(x+t)=(F.x)")) implies F is (2*t)-periodic & F is periodic
  proof
assume that
A1: t<>0 and
A2: for x st x in dom F holds (x+t in dom F & x-t in dom F & F.(x+t)=(F.x)");
 for x st x in dom F holds (x+2*t in dom F & x-2*t in dom F) & F.x=F.(x+2*t)
  proof
  let x;
  assume
A3:x in dom F; then
A4: x+t in dom F & x-t in dom F by A2; then
A5: x+t+t in dom F & x-t-t in dom F by A2;
   F.(x+2*t)=F.(x+t+t) .=(F.(x+t))" by A2,A4
               .=((F.x)")" by A2,A3
               .=F.x;
  hence thesis by A5;
  end;
then F is (2*t)-periodic by A1,Th1;
hence thesis;
end;
