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

  rr for set,

  rseq for Real_Sequence;

theorem Th41:
  rng(sec | [.0,PI/4.]) = [.1,sqrt 2.]
proof
  now
    let y be object;
    thus y in [.1,sqrt 2.] implies
ex x be object st x in dom (sec | [.0,PI/4.])
    & y = (sec | [.0,PI/4.]).x
    proof
      assume
A1:   y in [.1,sqrt 2.];
      then reconsider y1=y as Real;
      [.0,PI/4.] c= [.0,PI/2.[ by Lm5,XXREAL_2:def 12;
      then
A2:   sec|[.0,PI/4.] is continuous by Th37,FCONT_1:16;
      y1 in [.sec.0,sec.(PI/4).] \/ [.sec.(PI/4),sec.0.] by A1,Th31,
XBOOLE_0:def 3;
      then consider x be Real such that
A3:   x in [.0,PI/4.] & y1 = sec.x by A2,Lm13,Th1,FCONT_2:15,XBOOLE_1:1;
      take x;
      thus thesis by A3,Lm29,FUNCT_1:49;
    end;
    thus (ex x be object
st x in dom (sec | [.0,PI/4.]) & y = (sec | [.0,PI/4.]).
    x) implies y in [.1,sqrt 2.]
    proof
      given x be object such that
A4:   x in dom (sec | [.0,PI/4.]) and
A5:   y = (sec | [.0,PI/4.]).x;
      reconsider x1=x as Real by A4;
      y = sec.x1 by A4,A5,Lm29,FUNCT_1:49;
      hence thesis by A4,Lm29,Th33;
    end;
  end;
  hence thesis by FUNCT_1:def 3;
end;
