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

  rr for set,

  rseq for Real_Sequence;

theorem Th31:
  sec.0 = 1 & sec.(PI/4) = sqrt 2 & sec.(3/4*PI) = -sqrt 2 & sec. PI = -1
proof
A1: sec.PI = 1/(-1) by Lm6,Th2,RFUNCT_1:def 2,SIN_COS:76
    .= -1;
A2: sec.(3/4*PI) = 1/(-1/sqrt 2) by Lm6,Th2,Th30,RFUNCT_1:def 2
    .= -1/(1/sqrt 2) by XCMPLX_1:188
    .= -sqrt 2;
A3: sec.0 = 1/1 by Lm5,Th1,RFUNCT_1:def 2,SIN_COS:30
    .= 1;
  sec.(PI/4) = 1/(1/sqrt 2) by Lm5,Th1,Th29,RFUNCT_1:def 2
    .= sqrt 2;
  hence thesis by A3,A2,A1;
end;
