reserve x for Real;

theorem Th18:
  0 <= x & x < 2*PI & cos x = 0 implies x = PI/2 or x = 3/2*PI
proof
  assume that
A1: 0 <= x and
A2: x < 2*PI and
A3: cos x = 0;
A4: cos.x = 0 by A3,SIN_COS:def 19;
  then not x in ].PI/2,3/2*PI.[ by Th13;
  then
A5: PI/2 >= x or x >= 3/2*PI by XXREAL_1:4;
  not x in ].-PI/2,PI/2.[ by A4,Th11;
  then -PI/2 >= x or x >= PI/2 by XXREAL_1:4;
  then x = PI/2 or x = 3/2*PI or x > 3/2* PI by A1,A5,Lm1,XXREAL_0:1;
  then x = PI/2 or x = 3/2*PI or x in ].3/2* PI,2*PI.[ by A2,XXREAL_1:4;
  hence thesis by A4,Th15;
end;
