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

  rr for set,

  rseq for Real_Sequence;

theorem Th77:
  arcsec1|(sec.:[.0,PI/2.[) is increasing
proof
  set f = sec | [.0,PI/2.[;
A1: f.:[.0,PI/2.[ = rng(f|[.0,PI/2.[) by RELAT_1:115
    .= rng(sec|[.0,PI/2.[) by RELAT_1:73
    .= sec.:[.0,PI/2.[ by RELAT_1:115;
  f|[.0,PI/2.[ = f by RELAT_1:73;
  hence thesis by A1,Th17,FCONT_3:9;
end;
