reserve n,m,k for Nat,
  x,X for set,
  A for Subset of X,
  A1,A2 for SetSequence of X;

theorem Th73:
  A1 is convergent & A2 is convergent implies lim_sup (A1 (\+\) A2
  ) = lim_sup A1 \+\ lim_sup A2
proof
  assume that
A1: A1 is convergent and
A2: A2 is convergent;
  thus lim_sup (A1 (\+\) A2) = lim_sup ((A1 (\) A2) (\/) (A2 (\) A1)) by Th3
    .= lim_sup (A1 (\) A2) \/ lim_sup (A2 (\) A1) by Th68
    .= (lim_sup A1 \ lim_sup A2) \/ lim_sup (A2 (\) A1) by A2,Th72
    .= lim_sup A1 \+\ lim_sup A2 by A1,Th72;
end;
