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;
