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

theorem
  A1 is convergent implies (A (\) A1) is convergent & lim (A (\) A1) = A
  \ lim A1
proof
  assume
A1: A1 is convergent;
  then
A2: lim_inf (A (\) A1) = A \ lim_inf A1 by Th79
    .= A \ lim A1 by A1,KURATO_0:def 5;
  then lim_sup (A (\) A1) = lim_inf (A (\) A1) by A1,Th86;
  hence thesis by A2,KURATO_0:def 5;
end;
