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

  rr for set,

  rseq for Real_Sequence;

theorem Th79:
  arccosec1|(cosec.:[.-PI/2,0.[) is decreasing
proof
  set f = cosec | [.-PI/2,0.[;
A1: f.:[.-PI/2,0.[ = rng(f|[.-PI/2,0.[) by RELAT_1:115
    .= rng(cosec|[.-PI/2,0.[) by RELAT_1:73
    .= cosec.:[.-PI/2,0.[ by RELAT_1:115;
  f|[.-PI/2,0.[ = f by RELAT_1:73;
  hence thesis by A1,Th19,FCONT_3:10;
end;
