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 Th59:
  for p,q being Real holds cos.(p+q)-cos.p=
  -q*sin.p -q*Im((Sum(q*<i> P_dt))* (cos.p+(sin.p)*<i>))
proof
  let p,q be Real;
 cos.(p+q)-cos.p=cos.(p+q)-Re(Sum(p*<i> ExpSeq)) by Def18
    .=Re(Sum((p+q)*<i> ExpSeq))-Re(Sum(p*<i> ExpSeq)) by Def18
    .=Re(Sum((p*<i>+q*<i>) ExpSeq)-Sum(p*<i> ExpSeq)) by COMPLEX1:19
    .=Re((Sum(p*<i> ExpSeq))*(q*<i>)+q*<i>*(Sum((q*<i>) P_dt))*
  (Sum(p*<i> ExpSeq)))
  by Th58
    .=Re((cos.p+(sin.p)*<i>)*(q*<i>)
  +q*<i>*(Sum(q*<i> P_dt))*(Sum(p*<i> ExpSeq))) by Lm3
    .= Re((cos.p+(sin.p)*<i>)*(q*<i>)
  +q*<i>*(Sum(q*<i> P_dt))* (cos.p+(sin.p)*<i>)) by Lm3
    .=Re(-q*sin.p+q*cos.p*<i>)+Re(q*<i>*(Sum(q*<i> P_dt))*
  (cos.p+(sin.p)*<i>)) by COMPLEX1:8
    .=-q*sin.p+ Re(q*<i>*((Sum(q*<i> P_dt))* (cos.p+(sin.p)*<i>)))
  by COMPLEX1:12
    .=-q*sin.p -q*Im((Sum(q*<i> P_dt))* (cos.p+(sin.p)*<i>)) by Lm12;
  hence thesis;
end;
