reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;

theorem Th5:
  for n be Ordinal, A be finite Subset of n holds
    RelIncl n linearly_orders A
proof
   let X be Ordinal, A be finite Subset of X;
   set b=chi(A,X);
A1: support b c= A
   proof
     let y be object such that
A2:  y in support b;
A3:  b.y <>0 by A2,PRE_POLY:def 7;
     then y in dom b by FUNCT_1:def 2;
     hence thesis by A3,FUNCT_3:def 3;
   end;
A4: A c= support b
   proof
     let y be object such that
A5:  y in A;
     b.y = 1 by FUNCT_3:def 3,A5;
     hence thesis by PRE_POLY:def 7;
   end;
   reconsider b as bag of X by PRE_POLY:def 8,A1;
   RelIncl X linearly_orders support b by PRE_POLY:82;
   hence thesis by A4,A1,XBOOLE_0:def 10;
end;
