reserve x for Real;

theorem Th16:
  x in [.3/2*PI,2*PI.] implies cos.x >= 0
proof
  assume x in [.3/2*PI,2*PI.];
  then 3/2*PI <= x & x <= 2*PI by XXREAL_1:1;
  then x = 3/2*PI or x = 2*PI or 3/2*PI < x & x < 2*PI by XXREAL_0:1;
  then x = 3/2*PI or x = 2*PI or x in ].3/2*PI,2*PI.[ by XXREAL_1:4;
  hence thesis by Th15,SIN_COS:76;
end;
