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

  rr for set,

  rseq for Real_Sequence;

theorem Th69:
  0 <= r & r < PI/2 implies arcsec1 sec.r = r
proof
A1: dom (sec | [.0,PI/2.[) = [.0,PI/2.[ by Th1,RELAT_1:62;
  assume 0 <= r & r < PI/2;
  then
A2: r in [.0,PI/2.[;
  then arcsec1 sec.r = arcsec1.((sec|[.0,PI/2.[).r) by FUNCT_1:49
    .= (id [.0,PI/2.[).r by A2,A1,Th65,FUNCT_1:13
    .= r by A2,FUNCT_1:18;
  hence thesis;
end;
