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 Th62:
  for L be doubleLoopStr
    for f be finite Function, E be Function holds
      SgnMembershipNumber(f,L,E) = 1.L or
      SgnMembershipNumber(f,L,E) = -1.L
proof
  let L be doubleLoopStr;
  let f be finite Function, E be Function;
  set X={x where x is Element of dom f: x in dom f & f.x in E.x};
  X c= dom f
  proof
    let y be object;
    assume y in X;
    then ex x be Element of dom f st x=y & x in dom f & f.x in E.x;
    hence thesis;
  end;
  then reconsider X as finite set;
  card X is even or card X is odd;
  hence thesis by Def9;
end;
