reserve q,th,r for Real,
  a,b,p for Real,
  w,z for Complex,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,cq1 for Complex_Sequence,
  rseq,rseq1,rseq2 for Real_Sequence,
  rr for set,
  hy1 for 0-convergent non-zero Real_Sequence;
reserve d for Real;
reserve th,th1,th2 for Real;

theorem Th79:
  th in ].0,PI/2.[ implies cos.th > 0
proof
  assume that
A1: th in ].0,PI/2.[ and
A2: cos.th <= 0;
 cos|REAL is continuous by Th66,FDIFF_1:25;
then A3: cos|[.0,th.] is continuous by FCONT_1:16;
A4: 0 < th by A1,XXREAL_1:4;
   [.
cos.0,cos.th .] \/ [.cos.th,cos.0 .] = [.cos.th,cos.0 .] & 0 in [.cos.th,
  cos.0 .] by A2,Th30,XXREAL_1:1,222;
then  ex th2 st th2 in [.0,th .] & cos.th2 = 0
  by A3,A4,Th24,FCONT_2:15;
  then consider th2 be Real such that
A5: th2 in [.0,th .] and 0 < th and
A6: cos.th2 = 0 by A4;
A7: 0 <= th2 by A5,XXREAL_1:1;
A8: th2 <= th by A5,XXREAL_1:1;
A9: th < PI/2 by A1,XXREAL_1:4;
A10: 0<th2 by A6,A7,Th30;
 th2<PI/2 by A8,A9,XXREAL_0:2;
then A11: th2/2 < PI/2/2 by XREAL_1:74;
 PI in ].0, 4.[ by Def28;
then  PI < 4 by XXREAL_1:4;
then  PI/4 < 4/4 by XREAL_1:74;
then A12: th2/2 < 1 by A11,XXREAL_0:2;
 0 = cos.(th2/2 + th2/2) by A6
    .= (cos.(th2/2))^2 - (sin.(th2/2)) * (sin.(th2/2)) by Th73
    .= ((cos.(th2/2)) - (sin.(th2/2))) * ((cos.(th2/2)) + (sin.(th2/2)));
then
A13: cos.(th2/2) - sin.(th2/2) = 0 or cos.(th2/2) + sin.(th2/2) = 0;
A14: th2/2 in ].0,1 .[ by A10,A12,XXREAL_1:4;
 ].0,1 .[ c= [.0,1 .] by XXREAL_1:25;
then A15: cos.(th2/2) > 0 & sin.(th2/2) >= -0 by A14,Lm17,Th68;
 4*(th2/2) < 4*1 by A12,XREAL_1:68;
then A16: (2*th2) in ].0,4.[ by A10,XXREAL_1:4;
 (sin.(th2/2)) * (cos.(th2/2))"=1 by A13,A15,XCMPLX_0:def 7;
then  tan.((2*th2)/4) = 1 by A14,Th69,RFUNCT_1:def 1;
then  2*th2 =PI by A16,Def28;
  hence contradiction by A1,A8,XXREAL_1:4;
end;
