reserve r,r1,r2,g,g1,g2,x0,t for Real;
reserve n,k,m for Element of NAT;
reserve seq for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem
  f1 is_convergent_in x0 & f2 is_convergent_in x0 & (for r1,r2 st r1<x0
  & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom(f1-f2) & g2<r2 & x0<g2 & g2 in
dom(f1-f2)) implies f1-f2 is_convergent_in x0 & lim(f1-f2,x0)=(lim(f1,x0))-(lim
  (f2,x0))
proof
  assume that
A1: f1 is_convergent_in x0 and
A2: f2 is_convergent_in x0 and
A3: for r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom(f1-
  f2) & g2<r2 & x0<g2 & g2 in dom(f1-f2);
A4: -f2 is_convergent_in x0 by A2,Th32;
  hence f1-f2 is_convergent_in x0 by A1,A3,Th33;
  thus lim(f1-f2,x0)=lim(f1,x0)+lim(-f2,x0) by A1,A3,A4,Th33
    .=(lim(f1,x0))+-lim(f2,x0) by A2,Th32
    .=(lim(f1,x0))-lim(f2,x0);
end;
