reserve x for Real;

theorem Th9:
  x in ].PI,2*PI.[ implies sin.x < 0
proof
A1: sin.(x-PI) = sin.(-(PI-x)) .= -sin.(PI+-x) by SIN_COS:30
    .= --sin.(-x) by SIN_COS:78
    .= -sin.x by SIN_COS:30;
  assume
A2: x in ].PI,2*PI.[;
  then x < 2*PI by XXREAL_1:4;
  then
A3: x-PI < 2*PI-PI by XREAL_1:9;
  PI < x by A2,XXREAL_1:4;
  then PI-PI < x-PI by XREAL_1:9;
  then x-PI in ].0,PI.[ by A3,XXREAL_1:4;
  hence thesis by A1,Th7;
end;
