reserve x,x0, r, s, h for Real,

  n for Element of NAT,
  rr, y for set,
  Z for open Subset of REAL,

  f, f1, f2 for PartFunc of REAL,REAL;

theorem Th8:
  cot|].0,PI.[ is decreasing
proof
A1: for x st x in ].0,PI.[ holds diff(cot,x) < 0
  proof
    let x;
    assume
A2: x in ].0,PI.[;
    then 0 < sin.x by COMPTRIG:7;
    then (sin.x)^2 > 0;
    then 1/(sin.x)^2 > 0 /(sin.x)^2;
    then -1/(sin.x)^2 < -0;
    hence thesis by A2,Lm4;
  end;
  ].0,PI.[ c= dom cot by Lm2,FDIFF_1:def 6;
  hence thesis by A1,Lm2,ROLLE:10;
end;
