reserve x for Real;

theorem Th12:
  x in [.-PI/2,PI/2.] implies cos.x >= 0
proof
  assume x in [.-PI/2,PI/2.];
  then -PI/2 <= x & x <= PI/2 by XXREAL_1:1;
  then x = -PI/2 or x = PI/2 or -PI/2 < x & x < PI/2 by XXREAL_0:1;
  then x = -PI/2 or x = PI/2 or x in ].-PI/2,PI/2.[ by XXREAL_1:4;
  hence thesis by Th11,SIN_COS:30,76;
end;
