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 Th14:
  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))
   proof
   let x;
       assume
    x in dom F; then
    x+t in dom F & x-t in dom F by A1,Th1;
      hence thesis by A1,Th13;
      end;
   hence thesis by A2,Th1;
end;
