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 Th4:
  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:def 4;
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,Th1,A5;
        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:def 4;
        (F (#) G).x=F.x * G.x by A4,VALUED_1:def 4
                 .=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:def 4;
          hence thesis by A8,VALUED_1:def 4;
      end;
 hence thesis by A3,Th1;
end;
