reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;

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;
  thus lim (seq+seq1)=(lim seq)+(lim seq1) by A1,A2,SEQ_2:6
    .=(seq.0)+(lim seq1) by A1,Th25;
  thus lim (seq-seq1)=(lim seq)-(lim seq1) by A1,A2,SEQ_2:12
    .=(seq.0)-(lim seq1) by A1,Th25;
  thus lim (seq1-seq)=(lim seq1)-(lim seq) by A1,A2,SEQ_2:12
    .=(lim seq1)-(seq.0) by A1,Th25;
  thus lim (seq(#)seq1)=(lim seq)*(lim seq1) by A1,A2,SEQ_2:15
    .=(seq.0)*(lim seq1) by A1,Th25;
end;
