reserve N for Nat;
reserve n,m,n1,n2 for Nat;
reserve q,r,r1,r2 for Real;
reserve x,y for set;
reserve w,w1,w2,g,g1,g2 for Point of TOP-REAL N;
reserve seq,seq1,seq2,seq3,seq9 for Real_Sequence of N;

theorem
  seq is convergent & seq9 is convergent implies
  lim(seq - seq9) = (lim seq)-(lim seq9)
proof
  assume that
A1: seq is convergent and
A2: seq9 is convergent;
  - seq9 is convergent by A2,Th40;
  hence lim(seq - seq9)=(lim seq)+(lim(- seq9)) by A1,Th37
    .=(lim seq)+-(lim seq9) by A2,Th41
    .=(lim seq)-(lim seq9);
end;
