
theorem
  for n being Ordinal, T being connected TermOrder of n, b1,b2 being bag
  of n holds min(b1,b2,T) <= b1,T & min(b1,b2,T) <= b2,T
proof
  let n be Ordinal, T be connected TermOrder of n, b1,b2 be bag of n;
  per cases by Lm5;
  suppose
A1: b1 <= b2,T;
    then min(b1,b2,T) = b1 by Def4;
    hence thesis by A1,Lm2;
  end;
  suppose
A2: b2 <= b1,T;
    now
      per cases;
      case
A3:     b1 = b2;
        then min(b1,b2,T) = b1 by Lm6;
        hence thesis by A3,Lm2;
      end;
      case
        b1 <> b2;
        then b2 < b1,T by A2;
        then not b1 <= b2,T by Th5;
        then min(b1,b2,T) = b2 by Def4;
        hence thesis by A2,Lm2;
      end;
    end;
    hence thesis;
  end;
end;
