reserve r,r1,r2, s,x for Real,
  i for Integer;

theorem Th21:
  2*PI*i <= r & r < 2*PI+2*PI*i & sin r = 0 implies r = 2*PI*i or r = PI+2*PI*i
proof
  assume
A1: T(i) <= r & r < 2*PI+T(i);
  assume
A2: sin r = 0;
  then
A3: r <= PI+T(i) or 2*PI+T(i) <= r by Th12;
  r <= T(i) or r >= PI+T(i) by A2,Th11;
  hence thesis by A1,A3,XXREAL_0:1;
end;
