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

theorem Th22:
  2*PI*i <= r & r < 2*PI+2*PI*i & cos r = 0 implies r = PI/2+2*PI*
  i or r = 3/2*PI+2*PI*i
proof
  assume that
A1: T(i) <= r and
A2: r < 2*PI+T(i);
A3: -PI/2+T(i) < 0+T(i) by XREAL_1:6;
  assume
A4: cos r = 0;
  then
A5: PI/2+T(i) >= r or r >= 3/2*PI+T(i) by Th14;
  -PI/2+T(i) >= r or r >= PI/2+T(i) by A4,Th13;
  then r = PI/2+T(i) or r = 3/2*PI+T(i) or r > 3/2*PI+T(i) by A1,A5,A3,
XXREAL_0:1,2;
  hence thesis by A2,A4,Th15;
end;
