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
  sin^2 is (PI*i)-periodic
  proof
    i in INT by INT_1:def 2;
    then consider k being Nat such that
A1: i = k or i = - k by INT_1:def 1;
    per cases by A1;
    suppose i = k;
      then ex m being Nat st i = m + 1 by NAT_1:6;
      hence thesis by Lm22;
    end;
    suppose
A2:   i = -k;
      then consider m being Nat such that
A3:   -i = m + 1 by NAT_1:6;
      sin^2 is PI*(m+1)-periodic by Lm22;
      then sin^2 is -PI*(m+1)-periodic by Th14;
      hence thesis by A2,A3;
    end;
  end;
