reserve n,n1,m,m1,k for Nat;
reserve x,X,X1 for set;
reserve g,g1,g2,t,x0,x1,x2 for Complex;
reserve s1,s2,q1,seq,seq1,seq2,seq3 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f,f1,f2,h,h1,h2 for PartFunc of COMPLEX,COMPLEX;
reserve p,r,s for Real;
reserve Ns,Nseq for increasing sequence of NAT;

theorem
  seq is constant & seq1 is convergent implies lim (seq+seq1) =(seq.0) +
lim seq1 & lim (seq-seq1) =(seq.0) - lim seq1 & lim (seq1-seq) =(lim seq1) -seq
  .0 & lim (seq(#)seq1) =(seq.0) * (lim seq1)
proof
  assume that
A1: seq is constant and
A2: seq1 is convergent;
A3: seq is convergent by A1,Th26;
  hence lim (seq+seq1) =(lim seq)+(lim seq1) by A2,COMSEQ_2:14
    .=(seq.0)+(lim seq1) by A1,Th27;
  thus lim (seq-seq1) =(lim seq)-(lim seq1) by A2,A3,COMSEQ_2:26
    .=(seq.0)-(lim seq1) by A1,Th27;
  thus lim (seq1-seq) =(lim seq1)-(lim seq) by A2,A3,COMSEQ_2:26
    .=(lim seq1)-(seq.0) by A1,Th27;
  thus lim (seq(#)seq1) =(lim seq)*(lim seq1) by A2,A3,COMSEQ_2:30
    .=(seq.0)*(lim seq1) by A1,Th27;
end;
