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

  rr for set,

  rseq for Real_Sequence;

theorem Th48:
  [.1,sqrt 2.] c= dom arccosec2
proof
A1: [.PI/4,PI/2.] c= ].0,PI/2.] by Lm8,XXREAL_2:def 12;
  rng(cosec | [.PI/4,PI/2.]) c= rng(cosec | ].0,PI/2.])
  proof
    let y be object;
    assume y in rng(cosec | [.PI/4,PI/2.]);
    then y in cosec.:[.PI/4,PI/2.] by RELAT_1:115;
    then ex x be object
st x in dom cosec & x in [.PI/4,PI/2.] & y = cosec.x by
FUNCT_1:def 6;
    then y in cosec.:].0,PI/2.] by A1,FUNCT_1:def 6;
    hence thesis by RELAT_1:115;
  end;
  hence thesis by Th44,FUNCT_1:33;
end;
