reserve n,n1,n2,m for Nat,
  r,r1,r2,p,g1,g2,g for Real,
  seq,seq9,seq1 for Real_Sequence,
  y for set;

theorem Th12:
  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;
  thus lim(seq - seq9)=(lim seq)+(lim(- seq9)) by A1,A2,Th6
    .=(lim seq)+-(lim seq9) by A2,Th10
    .=(lim seq)-(lim seq9);
end;
