
theorem
  for n being Ordinal, T being connected TermOrder of n, b1,b2 being bag
  of n holds min(b1,b2,T) = b1 iff max(b1,b2,T) = b2
proof
  let n being Ordinal, T being connected TermOrder of n, b1,b2 being bag of n;
A1: now
    assume
A2: max(b1,b2,T) = b2;
    now
      per cases by A2,Def5;
      case
        not b2 <= b1,T;
        then b1 <= b2,T by Lm5;
        hence min(b1,b2,T) = b1 by Def4;
      end;
      case
        b1 = b2;
        hence min(b1,b2,T) = b1 by Th11;
      end;
    end;
    hence min(b1,b2,T) = b1;
  end;
  now
    assume
A3: min(b1,b2,T) = b1;
    now
      per cases by A3,Def4;
      case
        b1 <= b2,T;
        then max(b2,b1,T) = b2 by Def5;
        hence max(b1,b2,T) = b2 by Th15;
      end;
      case
        b1 = b2;
        hence max(b1,b2,T) = b2 by Th12;
      end;
    end;
    hence max(b1,b2,T) = b2;
  end;
  hence thesis by A1;
end;
