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

  rr for set,

  rseq for Real_Sequence;

theorem Th81:
  arcsec1|[.1,sqrt 2.] is increasing
proof
  [.1,sqrt 2.] = sec.:[.0,PI/4.] & [.0,PI/4.] c= [.0,PI/2.[ by Lm5,Th41,
RELAT_1:115,XXREAL_2:def 12;
  hence thesis by Th77,RELAT_1:123,RFUNCT_2:28;
end;
