
theorem
  for X being set, L being well-unital non empty multLoopStr_0 holds (
  1.L) |(X,L) = 1_(X,L)
proof
  let X be set, L be well-unital non empty multLoopStr_0;
  set o1 = (1.L) |(X,L), o2 = 1_(X,L);
  now
    set m = 0_(X,L)+*(EmptyBag X,1.L);
    let x be object;
    reconsider m as Function of Bags X, the carrier of L;
    reconsider m as Function of Bags X, L;
    reconsider m as Series of X, L;
    assume x in Bags X;
    then reconsider x9 = x as bag of X;
A1: dom(0_(X,L)) = dom((Bags X) --> 0.L) by POLYNOM1:def 8
      .= Bags X;
    then
A2: m = 0_(X,L)+*(EmptyBag X .--> 1.L) by FUNCT_7:def 3;
A3: EmptyBag X in dom(EmptyBag X .--> 1.L) by TARSKI:def 1;
A4: m.(EmptyBag X) = (0_(X,L)+*(EmptyBag X .--> 1.L)).(EmptyBag X) by A1,
FUNCT_7:def 3
      .= (EmptyBag X .--> 1.L).(EmptyBag X) by A3,FUNCT_4:13
      .= 1.L by FUNCOP_1:72;
    per cases;
    suppose
      x9 = EmptyBag X;
      hence o1.x = o2.x by A4,POLYNOM1:25;
    end;
    suppose
A5:   x9 <> EmptyBag X;
      then not x9 in dom(EmptyBag X .--> 1.L) by TARSKI:def 1;
      then m.x9 = (0_(X,L)).x9 by A2,FUNCT_4:11
        .= 0.L by POLYNOM1:22
        .= o2.x9 by A5,POLYNOM1:25;
      hence o1.x = o2.x;
    end;
  end;
  hence thesis;
end;
