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 Th1:
  F is t-periodic iff 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
 thus F is t-periodic implies 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
    assume
A1: F is t-periodic;
  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+t in dom F;
    hence thesis by A1;
  end;
  hence thesis by A1;
  end;
   assume
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);
   for x holds (x in dom F iff x+t in dom F) &
               (x in dom F implies F.x = F.(x+t))
   proof
     let x;
     x in dom F iff x+t in dom F
     proof
       x+t in dom F implies x in dom F
       proof
         assume x+t in dom F;
         then x+t-t in dom F by A2;
         hence thesis;
       end;
       hence thesis by A2;
     end;
     hence thesis by A2;
   end;
   hence thesis by A2;
 end;
