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

theorem Th11:
  2*PI*i < r & r < PI+2*PI*i implies sin r > 0
proof
  assume T(i) < r & r < PI+T(i);
  then T(i)-T(i) < r-T(i) & r-T(i) < PI+T(i)-T(i) by XREAL_1:9;
  then r-T(i) in ].0,PI.[ by XXREAL_1:4;
  then sin.(r+T(-i)) > 0 by COMPTRIG:7;
  then sin.r > 0 by Th8;
  hence thesis by SIN_COS:def 17;
end;
