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

  rr for set,

  rseq for Real_Sequence;

theorem Th80:
  arccosec2|(cosec.:].0,PI/2.]) is decreasing
proof
  set f = cosec | ].0,PI/2.];
A1: f.:].0,PI/2.] = rng(f|].0,PI/2.]) by RELAT_1:115
    .= rng(cosec|].0,PI/2.]) by RELAT_1:73
    .= cosec.:].0,PI/2.] by RELAT_1:115;
  f|].0,PI/2.] = f by RELAT_1:73;
  hence thesis by A1,Th20,FCONT_3:10;
end;
