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

  rr for set,

  rseq for Real_Sequence;

theorem Th52:
  arccosec2 | [.1,sqrt 2.] = (cosec | [.PI/4,PI/2.])"
proof
  set h = cosec | ].0,PI/2.];
A1: [.PI/4,PI/2.] c= ].0,PI/2.] by Lm8,XXREAL_2:def 12;
  then (cosec | [.PI/4,PI/2.])" = (h | [.PI/4,PI/2.])" by RELAT_1:74
    .= h" | (h.:[.PI/4,PI/2.]) by RFUNCT_2:17
    .= h" | rng (h | [.PI/4,PI/2.]) by RELAT_1:115
    .= h" | ([.1,sqrt 2.]) by A1,Th44,RELAT_1:74;
  hence thesis;
end;
