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

  rr for set,

  rseq for Real_Sequence;

theorem Th72:
  0 < r & r <= PI/2 implies arccosec2 cosec.r = r
proof
A1: dom (cosec | ].0,PI/2.]) = ].0,PI/2.] by Th4,RELAT_1:62;
  assume 0 < r & r <= PI/2;
  then
A2: r in ].0,PI/2.];
  then arccosec2 cosec.r = arccosec2.((cosec|].0,PI/2.]).r) by FUNCT_1:49
    .= (id ].0,PI/2.]).r by A2,A1,Th68,FUNCT_1:13
    .= r by A2,FUNCT_1:18;
  hence thesis;
end;
