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

  rr for set,

  rseq for Real_Sequence;

theorem Th71:
  -PI/2 <= r & r < 0 implies arccosec1 cosec.r = r
proof
A1: dom (cosec | [.-PI/2,0.[) = [.-PI/2,0.[ by Th3,RELAT_1:62;
  assume -PI/2 <= r & r < 0;
  then
A2: r in [.-PI/2,0.[;
  then arccosec1 cosec.r = arccosec1.((cosec|[.-PI/2,0.[).r) by FUNCT_1:49
    .= (id [.-PI/2,0.[).r by A2,A1,Th67,FUNCT_1:13
    .= r by A2,FUNCT_1:18;
  hence thesis;
end;
