reserve x for Real;

theorem Th10:
  x in [.PI,2*PI.] implies sin.x <= 0
proof
  assume x in [.PI,2*PI.];
  then PI <= x & x <= 2*PI by XXREAL_1:1;
  then x = PI or x = 2*PI or PI < x & x < 2*PI by XXREAL_0:1;
  then x = PI or x = 2*PI or x in ].PI,2*PI.[ by XXREAL_1:4;
  hence thesis by Th9,SIN_COS:76;
end;
