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

theorem
  cos|].3/2*PI,2*PI.[ is increasing
proof
  for th st th in ].3/2*PI,2*PI.[ holds diff(cos,(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(cos,(th)) = -sin.(PI + (PI/2+(th-3/2*PI))) by SIN_COS:67
      .= -(-sin.(PI/2+(th-3/2*PI))) by SIN_COS:78
      .= cos.(th-3/2*PI) by SIN_COS:78;
    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,SIN_COS:80;
  end;
  hence thesis by FDIFF_1:26,ROLLE:9,SIN_COS:24,67;
end;
