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

  rr for set,

  rseq for Real_Sequence;

theorem Th49:
  arcsec1 | [.1,sqrt 2.] = (sec | [.0,PI/4.])"
proof
  set h = sec | [.0,PI/2.[;
A1: [.0,PI/4.] c= [.0,PI/2.[ by Lm5,XXREAL_2:def 12;
  then (sec | [.0,PI/4.])" = (h | [.0,PI/4.])" by RELAT_1:74
    .= h" | (h.:[.0,PI/4.]) by RFUNCT_2:17
    .= h" | rng (h | [.0,PI/4.]) by RELAT_1:115
    .= h" | ([.1,sqrt 2.]) by A1,Th41,RELAT_1:74;
  hence thesis;
end;
