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

  rr for set,

  rseq for Real_Sequence;

theorem Th32:
  cosec.(-PI/2) = -1 & cosec.(-PI/4) = -sqrt 2 & cosec.(PI/4) =
  sqrt 2 & cosec.(PI/2) = 1
proof
A1: cosec.(PI/2) = 1/1 by Lm8,Th4,RFUNCT_1:def 2,SIN_COS:76
    .= 1;
A2: cosec.(PI/4) = 1/(1/sqrt 2) by Lm8,Th4,Th29,RFUNCT_1:def 2
    .= sqrt 2;
A3: cosec.(-PI/2) = 1/sin.(-PI/2) by Lm7,Th3,RFUNCT_1:def 2
    .= 1/(-1) by SIN_COS:30,76
    .= -1;
  cosec.(-PI/4) = 1/(-1/sqrt 2) by Lm7,Th3,Th30,RFUNCT_1:def 2
    .= -1/(1/sqrt 2) by XCMPLX_1:188
    .= -sqrt 2;
  hence thesis by A3,A2,A1;
end;
