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 Th60:
  for p,q being Real holds sin.(p+q)-sin.p=
  q*cos.p+q*Re((Sum((q*<i>) P_dt))* (cos.p+sin.p*<i>))
proof
  let p,q be Real;
 sin.(p+q)-sin.p=sin.(p+q)-Im(Sum((p*<i>) ExpSeq)) by Def16
    .=Im(Sum(((p+q)*<i>) ExpSeq))-Im(Sum((p*<i>) ExpSeq)) by Def16
    .=Im(Sum(((p*<i>)+(q*<i>)) ExpSeq)-Sum((p*<i>) ExpSeq)) by COMPLEX1:19
    .=Im((Sum((p*<i>) ExpSeq))*(q*<i>)
  +(q*<i>)*(Sum((q*<i>) P_dt))*(Sum((p*<i>) ExpSeq))) by Th58
    .=Im((cos.p+(sin.p)*<i>)*(q*<i>)
  +(q*<i>)*(Sum((q*<i>) P_dt))*(Sum((p*<i>) ExpSeq))) by Lm3
    .= Im((cos.p+sin.p*<i>)*(q*<i>)
  +(q*<i>)*(Sum((q*<i>) P_dt))* (cos.p+(sin.p)*<i>)) by Lm3
    .=Im(-q*sin.p+(q*cos.p)*<i>)+Im((q*<i>)*(Sum((q*<i>) P_dt))
  * (cos.p+sin.p*<i>)) by COMPLEX1:8
    .=q*cos.p+Im((q*<i>)*((Sum((q*<i>) P_dt))* (cos.p+sin.p*<i>)))
  by COMPLEX1:12
    .=q*cos.p+q*Re((Sum((q*<i>) P_dt))* (cos.p+sin.p*<i>)) by Lm12;
  hence thesis;
end;
