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 Th61:
  for p,q being Real holds exp_R.(p+q)-exp_R.p=
  q * (exp_R.p)+q*exp_R.p*Re((Sum(q P_dt)))
proof
  let p,q be Real;
 exp_R.(p+q)-exp_R.p=exp_R.(p+q)-Re(Sum(p ExpSeq)) by Th47
    .=Re(Sum((p+q) ExpSeq))-Re(Sum(p ExpSeq)) by Th47
    .=Re(Sum((p+q) ExpSeq)-Sum(p ExpSeq)) by COMPLEX1:19
    .=Re((Sum(p ExpSeq))*q
  +q*(Sum(q P_dt))*(Sum(p ExpSeq))) by Th58
    .=Re((Sum(p ExpSeq))*q)+
  Re(q*(Sum(q P_dt))*(Sum(p ExpSeq))) by COMPLEX1:8
    .=q*Re(Sum(p ExpSeq))+
  Re(q*(Sum(q P_dt))*(Sum(p ExpSeq))) by Lm12
    .=q* (exp_R.p)+ Re(q*((Sum(q P_dt))*(Sum(p ExpSeq))))
  by Th47
    .=q* (exp_R.p)+ q*Re((Sum(q P_dt))*(Sum(p ExpSeq))) by Lm12
    .=q* (exp_R.p)+ q*Re((Sum(q P_dt))*Sum(p rExpSeq)) by Lm9
    .=q* (exp_R.p)+q*Re((Sum(q P_dt))*(exp_R.p+0*<i>)) by Def22
    .=q* (exp_R.p)+q*(exp_R.p*Re((Sum(q P_dt)))) by Lm12
    .=q* (exp_R.p)+q*exp_R.p*Re((Sum(q P_dt)));
  hence thesis;
end;
