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
  (t1-t2<>0 & for x st x in dom F holds (x+t1 in dom F & x-t1 in dom F &
  x+t2 in dom F & x-t2 in dom F &
  F.(x+t1)=F.(x+t2))) implies F is (t1-t2)-periodic & F is periodic
proof
assume that
A1: t1-t2<>0 and
A2: for x st x in dom F holds (x+t1 in dom F & x-t1 in dom F &
 x+t2 in dom F & x-t2 in dom F & F.(x+t1)=F.(x+t2));
 for x st x in dom F
holds (x+(t1-t2) in dom F & x-(t1-t2) in dom F) & F.x=F.(x+(t1-t2))
  proof
  let x;
  assume x in dom F; then
A3: x+t1 in dom F & x-t1 in dom F & x+t2 in dom F & x-t2 in dom F by A2;
  then
A4: x+t1-t2 in dom F & x-t1+t2 in dom F by A2;
  F.(x+(t1-t2))=F.(x-t2+t1) .=F.(x-t2+t2) by A2,A3
               .=F.x;
  hence thesis by A4;
  end;
then F is (t1-t2)-periodic by Th1,A1;
hence thesis;
end;
