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

  rr for set,

  rseq for Real_Sequence;

theorem Th47:
  [.-sqrt 2,-1.] c= dom arccosec1
proof
A1: [.-PI/2,-PI/4.] c= [.-PI/2,0.[ by Lm7,XXREAL_2:def 12;
  rng(cosec | [.-PI/2,-PI/4.]) c= rng(cosec | [.-PI/2,0.[)
  proof
    let y be object;
    assume y in rng(cosec | [.-PI/2,-PI/4.]);
    then y in cosec.:[.-PI/2,-PI/4.] by RELAT_1:115;
    then ex x be object
st x in dom cosec & x in [.-PI/2,-PI/4.] & y = cosec.x by
FUNCT_1:def 6;
    then y in cosec.:[.-PI/2,0.[ by A1,FUNCT_1:def 6;
    hence thesis by RELAT_1:115;
  end;
  hence thesis by Th43,FUNCT_1:33;
end;
