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 Th72:
  sin.(PI/4) = cos.(PI/4)
proof
A1: PI in ].0, 4.[ by Def28;
then A2: 0< PI by XXREAL_1:4;
 PI < 4 by A1,XXREAL_1:4;
then  PI/4 < 4/4 by XREAL_1:74;
then A3: PI/4 in ].0,1 .[ by A2,XXREAL_1:4;
 tan.(PI/4) =1 by Def28;
then  sin.(PI/4) * (cos.(PI/4))"=1 by A3,Th69,RFUNCT_1:def 1;
  hence thesis by XCMPLX_1:209;
end;
