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 Th6:
  F is t-periodic implies -F is t-periodic
proof
  assume
A1: F is t-periodic;
    then
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) by Th1;
   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
A3:     x in dom F by VALUED_1:8;
        then
A4:     x+t in dom F & x-t in dom F by A1,Th1;
        (-F).x=-F.x by VALUED_1:8
                 .=-F.(x+t) by A1,A3
                 .=(-F).(x+t) by VALUED_1:8;
          hence thesis by A4,VALUED_1:8;
      end;
 hence thesis by A2,Th1;
end;
