reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;
reserve V, C for set;

theorem
  for n being Ordinal, b being bag of n holds RelIncl n
  linearly_orders support b
proof
  let n be Ordinal, b be bag of n;
  set R = RelIncl n, s = support b;
  for x,y being object holds x in s & y in s & x <> y implies [x,y] in R or [
  y,x] in R
  proof
    let x,y be object;
    assume that
A1: x in s and
A2: y in s and
    x <> y;
    assume
A3: not [x,y] in R;
    reconsider x,y as Ordinal by A1,A2;
    y c= x by A1,A2,A3,WELLORD2:def 1;
    hence thesis by A1,A2,WELLORD2:def 1;
  end;
  then
A4: R is_connected_in s;
A5: R is_antisymmetric_in s by Lm8;
A6: R is_transitive_in s by Lm8;
  R is_reflexive_in s by Lm8;
  hence thesis by A4,A5,A6,ORDERS_1:def 9;
end;
