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 Th28:
  (cos.th)^2+(sin.th)^2=1 & (cos.th)*(cos.th)+(sin.th)*(sin.th)=1
proof
  reconsider th1=th as Real;
A1: Sum((th1*<i>) ExpSeq ) * Sum((-(th1*<i>)) ExpSeq )
  =Sum(((th1*<i>)+(-(th1*<i>))) ExpSeq ) by Lm2
    .=1r by Th9;
  thus
 (cos.(th))^2+(sin.(th))^2
  =(Re(Sum((th1*<i>) ExpSeq )))^2+(sin.(th))^2 by Def18
    .=(Re(Sum((th1*<i>) ExpSeq )))^2+(Im(Sum((th1*<i>) ExpSeq )))^2 by Def16
    .=|.Sum((th1*<i>) ExpSeq )*Sum((th1*<i>) ExpSeq ).| by COMPLEX1:68
    .=|.Sum((th1*<i>) ExpSeq ) * Sum((th1*<i>) ExpSeq)*'.| by COMPLEX1:69
    .=1 by A1,Lm4,COMPLEX1:48;
  hence thesis;
end;
