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 Th30:
  cos.0=1 & sin.0=0 & cos.(-th)=cos.th & sin.(-th)=-sin.th
proof
  thus cos.0 = Re(Sum(0*<i> ExpSeq)) by Def18
    .= 1 by Th9,COMPLEX1:6;
  thus sin.0 = Im(Sum(0*<i> ExpSeq)) by Def16
    .= 0 by Th9,COMPLEX1:6;
  reconsider th1=th as Real;
  thus cos.(-th) = Re( Sum((-0+(-th1)*<i>) ExpSeq) ) by Def18
    .= Re( Sum((-(th1*<i>)) ExpSeq) )
    .= Re(Sum((th1*<i>) ExpSeq)*') by Lm4
    .= Re((cos.(th)+(sin.th)*<i>)*') by Lm3
    .= Re(cos.(th)+(-sin.th)*<i>) by Lm1
    .= cos.(th) by COMPLEX1:12;
  thus sin.(-th) = Im( Sum((-0+(-th1)*<i>) ExpSeq) ) by Def16
    .= Im( Sum((-(th1*<i>)) ExpSeq) )
    .= Im(Sum((th1*<i>) ExpSeq)*') by Lm4
    .= Im((cos.(th)+(sin.th)*<i>)*') by Lm3
    .= Im(cos.(th)+(-sin.th)*<i>) by Lm1
    .= -sin.(th) by COMPLEX1:12;
end;
