
theorem LM1:
  for X be RealUnitarySpace, Y be Subset of X holds
    Y is closed iff
    for s be sequence of X st rng s c= Y & s is convergent
      holds lim s in Y
proof
  let X be RealUnitarySpace, Y be Subset of X;
  hereby assume Y is closed; then
    consider Z be Subset of RUSp2RNSp X such that
A1:   Z = Y & Z is closed;
    thus for s be sequence of X st rng s c= Y & s is convergent
      holds lim s in Y
    proof
      let s be sequence of X;
      assume A3: rng s c= Y & s is convergent;
      reconsider s1=s as sequence of RUSp2RNSp X;
      rng s1 c= Z & s1 is convergent by A3,A1,RHS8; then
      lim s1 in Z by A1;
      hence lim s in Y by A1,A3,RHS9;
    end;
  end;
  assume A4: for s be sequence of X st rng s c= Y & s is convergent
               holds lim s in Y;
  reconsider Z=Y as Subset of RUSp2RNSp X;
  for s1 be sequence of RUSp2RNSp X st rng s1 c= Z & s1 is convergent
    holds lim s1 in Z
  proof
    let s1 be sequence of RUSp2RNSp X;
    assume A5: rng s1 c= Z & s1 is convergent;
    reconsider s=s1 as sequence of X;
A6: rng s c= Y & s is convergent by A5,RHS8; then
    lim s in Y by A4;
    hence lim s1 in Z by A6,RHS9;
  end; then
  Z is closed;
  hence Y is closed;
end;
