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;
