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

theorem
  cos|].PI,3/2*PI.[ is increasing
proof
  for th st th in ].PI,3/2*PI.[ holds diff(cos,(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;
A3: diff(cos,(th)) = -sin.(PI+(th-PI)) by SIN_COS:67
      .= -(-sin.(th-PI)) by SIN_COS:78
      .= sin.(th-PI);
    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;
    hence thesis by A3,Lm1;
  end;
  hence thesis by FDIFF_1:26,ROLLE:9,SIN_COS:24,67;
end;
