reserve X for Banach_Algebra,
  w,z,z1,z2 for Element of X,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,s,s9 for sequence of X,
  rseq for Real_Sequence;

theorem Th1:
  seq1 is convergent & seq2 is convergent & lim(seq1-seq2)=0.X
  implies lim seq1 = lim seq2
proof
  assume that
A1: seq1 is convergent and
A2: seq2 is convergent and
A3: lim(seq1-seq2)=0.X;
  lim(seq1)-lim(seq2) =0.X by A1,A2,A3,NORMSP_1:26;
  then lim(seq1)-lim(seq2) + lim(seq2) = lim(seq2) by LOPBAN_3:38;
  then lim(seq1)-(lim(seq2) - lim(seq2)) = lim(seq2) by LOPBAN_3:38;
  then lim(seq1)-0.X = lim(seq2) by RLVECT_1:15;
  hence thesis by RLVECT_1:13;
end;
