reserve x for Real;

theorem Th13:
  x in ].PI/2,3/2*PI.[ implies cos.x < 0
proof
A1: sin.(x+PI/2) = cos.x by SIN_COS:78;
  assume
A2: x in ].PI/2,3/2*PI.[;
  then x < 3/2*PI by XXREAL_1:4;
  then
A3: x+PI/2 < 3/2*PI+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 ].PI,2*PI.[ by A3,XXREAL_1:4;
  hence thesis by A1,Th9;
end;
