reserve i,j,k,n,m for Nat,
        b,b1,b2 for bag of n;

theorem  Th16:
  for O being Ordinal,
      A being finite Subset of Bags O st
     n in dom SgmX(BagOrder O, A) & m in dom SgmX(BagOrder O, A) & n < m
   holds SgmX(BagOrder O, A)/.n < SgmX(BagOrder O, A)/.m
proof
  let O be Ordinal,A be finite Subset of Bags O;
  set S=SgmX(BagOrder O, A);
  assume
A1: n in dom S & m in dom S & n < m;
  BagOrder O linearly_orders field BagOrder O by ORDERS_1:37;
  then BagOrder O linearly_orders Bags O by ORDERS_1:12;
  then BagOrder O linearly_orders A by ORDERS_1:38;
  then A2: S/.n <> S/.m & [S/.n,S/.m] in BagOrder O by PRE_POLY:def 2,A1;
  then S/.n <=' S/.m by PRE_POLY:def 14;
  hence thesis by A2,PRE_POLY:def 10;
end;
