reserve x for Real;

theorem Th7:
  x in ].0,PI.[ implies sin.x > 0
proof
  assume
A1: x in ].0,PI.[;
  per cases by A1,Th6;
  suppose
A2: x in ].0,PI/2.[;
    then x < PI/2 by XXREAL_1:4;
    then -x > -PI/2 by XREAL_1:24;
    then
A3: -x+PI/2 > -PI/2+PI/2 by XREAL_1:6;
    0 < x by A2,XXREAL_1:4;
    then -x+PI/2 < 0+PI/2 by XREAL_1:6;
    then PI/2-x in ].0,PI/2.[ by A3,XXREAL_1:4;
    then cos.(PI/2-x) > 0 by SIN_COS:80;
    hence thesis by SIN_COS:78;
  end;
  suppose
    x=PI/2;
    hence thesis by SIN_COS:76;
  end;
  suppose
A4: x in ].PI/2,PI.[;
    then x < PI by XXREAL_1:4;
    then
A5: x-PI/2 < PI-PI/2 by XREAL_1:9;
    PI/2 < x by A4,XXREAL_1:4;
    then PI/2-PI/2 < x-PI/2 by XREAL_1:9;
    then x-PI/2 in ].0,PI/2.[ by A5,XXREAL_1:4;
    then cos.(-(PI/2-x)) > 0 by SIN_COS:80;
    then cos.(PI/2-x) > 0 by SIN_COS:30;
    hence thesis by SIN_COS:78;
  end;
end;
