reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem
  f1 is convergent_in+infty & f2 is convergent_in+infty & (for r ex g st
r<g & g in dom(f1-f2)) implies f1-f2 is convergent_in+infty & lim_in+infty(f1-
  f2)=(lim_in+infty f1)-(lim_in+infty f2)
proof
  assume that
A1: f1 is convergent_in+infty and
A2: f2 is convergent_in+infty and
A3: for r ex g st r<g & g in dom(f1-f2);
A4: -f2 is convergent_in+infty by A2,Th81;
  hence f1-f2 is convergent_in+infty by A1,A3,Th82;
  lim_in+infty(-f2)=-(lim_in+infty f2) by A2,Th81;
  hence lim_in+infty(f1-f2)=lim_in+infty f1+-lim_in+infty f2 by A1,A3,A4,Th82
    .=lim_in+infty f1-lim_in+infty f2;
end;
