reserve x for Real;

theorem Th11:
  x in ].-PI/2,PI/2.[ implies cos.x > 0
proof
A1: sin.(x+PI/2) = cos.x by SIN_COS:78;
  assume
A2: x in ].-PI/2,PI/2.[;
  then x < PI/2 by XXREAL_1:4;
  then
A3: x+PI/2 < PI/2+PI/2 by XREAL_1:6;
  -PI/2 < x by A2,XXREAL_1:4;
  then -PI/2+PI/2 < x+PI/2 by XREAL_1:6;
  then x+PI/2 in ].0,PI.[ by A3,XXREAL_1:4;
  hence thesis by A1,Th7;
end;
