reserve x for Real;

theorem Th15:
  x in ].3/2*PI,2*PI.[ implies cos.x > 0
proof
A1: cos.(x-PI) = cos.(-(PI-x)) .= cos.(PI+-x) by SIN_COS:30
    .= -cos.(-x) by SIN_COS:78
    .= -cos.x by SIN_COS:30;
  assume
A2: x in ].3/2*PI,2*PI.[;
  then x < 2*PI by XXREAL_1:4;
  then x-PI < 2*PI-PI by XREAL_1:9;
  then
A3: x-PI < 3/2*PI by Lm5,XXREAL_0:2;
  3/2*PI < x by A2,XXREAL_1:4;
  then 3/2*PI-PI < x-PI by XREAL_1:9;
  then x-PI in ].PI/2,3/2*PI.[ by A3,XXREAL_1:4;
  hence thesis by A1,Th13;
end;
