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

  rr for set,

  rseq for Real_Sequence;

theorem Th51:
  arccosec1 | [.-sqrt 2,-1.] = (cosec | [.-PI/2,-PI/4.])"
proof
  set h = cosec | [.-PI/2,0.[;
A1: [.-PI/2,-PI/4.] c= [.-PI/2,0.[ by Lm7,XXREAL_2:def 12;
  then (cosec | [.-PI/2,-PI/4.])" = (h | [.-PI/2,-PI/4.])" by RELAT_1:74
    .= h" | (h.:[.-PI/2,-PI/4.]) by RFUNCT_2:17
    .= h" | rng (h | [.-PI/2,-PI/4.]) by RELAT_1:115
    .= h" | ([.-sqrt 2,-1.]) by A1,Th43,RELAT_1:74;
  hence thesis;
end;
