
theorem
  for n being Element of NAT, T being connected admissible TermOrder of
  n, P being non empty Subset of Bags n ex b being bag of n st b in P & for b9
  being bag of n st b9 in P holds b <= b9,T
proof
  let n be Element of NAT, T be connected admissible TermOrder of n, P be non
  empty Subset of Bags n;
  set R = RelStr(#Bags n, T#), m = MinElement(P,R);
A1: T is_connected_in field T by RELAT_2:def 14;
  reconsider b = m as bag of n;
A2: m is_minimal_wrt P,the InternalRel of R by BAGORDER:def 17;
A3: now
    let b9 be bag of n;
    b <= b,T by TERMORD:6;
    then [b,b] in T by TERMORD:def 2;
    then
A4: b in field T by RELAT_1:15;
    b9 <= b9,T by TERMORD:6;
    then [b9,b9] in T by TERMORD:def 2;
    then
A5: b9 in field T by RELAT_1:15;
    assume
A6: b9 in P;
    now
      per cases;
      case
        b9 = b;
        hence b <= b9,T by TERMORD:6;
      end;
      case
A7:     b9 <> b;
        then not [b9,b] in T by A2,A6,WAYBEL_4:def 25;
        then [b,b9] in T by A1,A4,A5,A7,RELAT_2:def 6;
        hence b <= b9,T by TERMORD:def 2;
      end;
    end;
    hence b <= b9,T;
  end;
  m in P by BAGORDER:def 17;
  hence thesis by A3;
end;
