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