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

theorem
  f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & (for r st
r<x0 ex g st r<g & g<x0 & g in dom(f1-f2)) implies f1-f2 is_left_convergent_in
  x0 & lim_left(f1-f2,x0)=(lim_left(f1,x0))-(lim_left(f2,x0))
proof
  assume that
A1: f1 is_left_convergent_in x0 and
A2: f2 is_left_convergent_in x0 and
A3: for r st r<x0 ex g st r<g & g<x0 & g in dom(f1-f2);
A4: -f2 is_left_convergent_in x0 by A2,Th44;
  hence f1-f2 is_left_convergent_in x0 by A1,A3,Th45;
  thus lim_left(f1-f2,x0)=lim_left(f1,x0)+lim_left(-f2,x0) by A1,A3,A4,Th45
    .=(lim_left(f1,x0))+-lim_left(f2,x0) by A2,Th44
    .=(lim_left(f1,x0))-lim_left(f2,x0);
end;
