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

  rr for set,

  rseq for Real_Sequence;

theorem Th113:
  1 <= r & r <= sqrt 2 implies sin.(arcsec1 r) = sqrt(r^2-1)/r &
  cos.(arcsec1 r) = 1/r
proof
  set x = arcsec1 r;
  assume that
A1: 1 <= r and
A2: r <= sqrt 2;
  r in [.1,sqrt 2.] by A1,A2;
  then
A3: x in dom (sec | [.0,PI/4.]) by Lm29,Th85;
  PI/4 < PI/1 by XREAL_1:76;
  then 0 in [.0,PI.] & PI/4 in [.0,PI.];
  then [.0,PI/4.] c= [.0,PI.] by XXREAL_2:def 12;
  then
A4: sin.x >= 0 by A3,Lm29,COMPTRIG:8;
A5: dom (sec | [.0,PI/4.]) c= dom sec by RELAT_1:60;
A6: r = (cos^).x by A1,A2,Th89
    .= 1/cos.x by A3,A5,RFUNCT_1:def 2;
  r^2 >= 1^2 by A1,SQUARE_1:15;
  then
A7: r^2-1 >= 0 by XREAL_1:48;
  (sin.x)^2+(cos.x)^2 = 1 by SIN_COS:28;
  then (sin.x)^2 = 1-(cos.x)^2 .= 1-(1/r)*(1/r) by A6
    .= 1-1/(r^2) by XCMPLX_1:102
    .= (r^2)/(r^2)-1/(r^2) by A1,XCMPLX_1:60
    .= (r^2-1)/(r^2);
  then sin.x = sqrt ((r^2-1)/(r^2)) by A4,SQUARE_1:def 2
    .= sqrt(r^2-1)/sqrt(r^2) by A1,A7,SQUARE_1:30
    .= sqrt(r^2-1)/r by A1,SQUARE_1:22;
  hence thesis by A6;
end;
