reserve p,q,r,th,th1 for Real;
reserve n for Nat;

theorem
  sin|].PI,3/2*PI.[ is decreasing
proof
  for th st th in ].PI,3/2*PI.[ holds diff(sin,(th))<0
  proof
    let th such that
A1: th in ].PI,3/2*PI.[;
    th < 3/2*PI by A1,XXREAL_1:4;
    then
A2: th-PI < 3/2*PI-PI by XREAL_1:9;
    PI < th by A1,XXREAL_1:4;
    then PI-PI < th-PI by XREAL_1:9;
    then th-PI in ].0,PI/2.[ by A2,XXREAL_1:4;
    then cos.(th-PI) > 0 by SIN_COS:80;
    then
A3: 0-cos.(th-PI) < 0;
    diff(sin,(th))= cos.(PI+(th-PI)) by SIN_COS:68
      .= -cos.(th-PI) by SIN_COS:78;
    hence thesis by A3;
  end;
  hence thesis by FDIFF_1:26,ROLLE:10,SIN_COS:24,68;
end;
