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 Th3:
  F is t-periodic & G is t-periodic implies F-G is t-periodic
proof
    assume that
A1: F is t-periodic and
A2: G is t-periodic;
A3: 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) by A1,Th1;
    for x st x in dom (F - G)
    holds (x+t in dom (F - G) & x-t in dom (F - G)) & (F - G).x=(F - G).(x+t)
    proof
      let x;
      assume
A4:   x in dom (F - G); then
A5:   x in dom F /\ dom G by VALUED_1:12;
A6:   dom F /\ dom G c= dom F & dom F /\ dom G c= dom G by XBOOLE_1:17;
      then
A7:   x+t in dom F & x-t in dom F by A1,A5,Th1;
      x+t in dom G & x-t in dom G by A2,Th1,A5,A6; then
A8:   x+t in dom F /\ dom G & x-t in dom F /\ dom G by A7,XBOOLE_0:def 4;
      then
A9:   x+t in dom (F - G) & x-t in dom (F - G) by VALUED_1:12;
      (F - G).x=F.x - G.x by A4,VALUED_1:13
                 .=F.(x+t)-G.x by A1,A5,A6
                 .=F.(x+t)-G.(x+t) by A2,A5,A6
                 .=(F - G).(x+t) by A9,VALUED_1:13;
      hence thesis by A8,VALUED_1:12;
    end;
    hence thesis by A3,Th1;
end;
