reserve x for Real;

theorem Th8:
  x in [.0,PI.] implies sin.x >= 0
proof
  assume
A1: x in [.0,PI.];
  then x <= PI by XXREAL_1:1;
  then x = 0 or x = PI or 0 < x & x < PI by A1,XXREAL_0:1,XXREAL_1:1;
  then x = 0 or x = PI or x in ].0,PI.[ by XXREAL_1:4;
  hence thesis by Th7,SIN_COS:30,76;
end;
