reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
          right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
             right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;

theorem Th56:
  for L be doubleLoopStr, E be Function holds
    SgnMembershipNumber({},L,E) = 1.L
proof
  let L be doubleLoopStr, E be Function;
  reconsider f={} as finite Function;
  set X = {x where x is Element of dom f: x in dom f & f.x in E.x};
A1: X={}
  proof
    assume X<>{};
    then consider y be object such that
A2: y in X by XBOOLE_0:def 1;
    ex x be Element of dom f st y =x & x in dom f & f.x in E.x by A2;
    hence thesis;
  end;
  then reconsider X as finite set;
  card X is even by A1;
  hence thesis by Def9;
end;
