reserve x,x0, r,r1,r2 for Real,
      th for Real,

  rr for set,

  rseq for Real_Sequence;

theorem
  cosec.:].0,PI/2.[ is open
proof
  for x0 st x0 in ].0,PI/2.[ holds diff(cosec,x0) < 0
  proof
    let x0;
    assume
A1: x0 in ].0,PI/2.[;
    ].0,PI/2.[ c= ].-PI/2,PI/2.[ by XXREAL_1:46;
    then
A2: cos.x0 > 0 by A1,COMPTRIG:11;
    ].0,PI/2.[ c= ].0,PI.[ by COMPTRIG:5,XXREAL_1:46;
    then sin.x0 > 0 by A1,COMPTRIG:7;
    then -(cos.x0/(sin.x0)^2) < -0 by A2;
    hence thesis by A1,Th8;
  end;
  then rng(cosec|].0,PI/2.[) is open by Lm18,Th8,FDIFF_2:41;
  hence thesis by RELAT_1:115;
end;
