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

theorem
  sin|].3/2*PI,2*PI.[ is increasing
proof
  for th st th in ].3/2*PI,2*PI.[ holds diff(sin,(th))>0
  proof
    let th such that
A1: th in ].3/2*PI,2*PI.[;
    th < 2*PI by A1,XXREAL_1:4;
    then
A2: th-3/2*PI < 2*PI-3/2*PI by XREAL_1:9;
A3: diff(sin,(th)) = cos.(PI + (PI/2+(th-3/2*PI))) by SIN_COS:68
      .= -cos.(PI/2+(th-3/2*PI)) by SIN_COS:78
      .= -(-sin.(th-3/2*PI)) by SIN_COS:78
      .= sin.(th-3/2*PI);
    3/2*PI < th by A1,XXREAL_1:4;
    then 3/2*PI-3/2*PI < th-3/2*PI by XREAL_1:9;
    then th-3/2*PI in ].0,PI/2.[ by A2,XXREAL_1:4;
    hence thesis by A3,Lm1;
  end;
  hence thesis by FDIFF_1:26,ROLLE:9,SIN_COS:24,68;
end;
