 reserve X,a,b,c,x,y,z,t for set;
 reserve R for Relation;

theorem Prop14: :: Proposition 14
  for A being non empty finite set,
      U being Function of bool A, bool A st
    U.{} = {} &
    (for X being Subset of A holds U.((U.X)`) c= (U.X)`) &
    (for X,Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y) holds
  ex R being negative_alliance finite non empty RelStr st
  the carrier of R = A & U = UAp R
  proof
    let A be non empty finite set;
    let U be Function of bool A,bool A;
    assume
a0: U.{} = {} &
    (for X being Subset of A holds U.((U.X)`) c= (U.X)`) &
    (for X,Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y); then
    consider R being non empty finite RelStr such that
a1: the carrier of R = A & LAp R = Flip U & UAp R = U &
      for x,y being Element of R holds
      [x,y] in the InternalRel of R iff x in U.{y} by YaoTh3;
    set X = the carrier of R;
    set I = the InternalRel of R;
    for x,y being object st x in X & y in X holds
    (ex z being object st z in X &
      [x,z] in I & not [z,y] in I) implies not [x,y] in I
    proof
      let x,y be object;
      assume x in X & y in X; then
      reconsider xx = x, yy = y as Element of R;
      given z being object such that
A1:   z in X & [x,z] in I & not [z,y] in I;
      reconsider zz = z as Element of R by A1;
      not zz in ((UAp R).{yy}) by A1,a1; then
B0:   zz in ((UAp R).{yy})` by XBOOLE_0:def 5;
B1:   zz in Class (the InternalRel of R, xx) by RELAT_1:169,A1;
B5:   (UAp R).(((UAp R).{yy})`) c= ((UAp R).{yy})` by a0,a1;
      Class (the InternalRel of R, xx) meets ((UAp R).{yy})`
        by B1,XBOOLE_0:3,B0; then
      xx in { x where x is Element of R :
        Class (the InternalRel of R, x) meets ((UAp R).{yy})` }; then
      xx in UAp (((UAp R).{yy})`) by ROUGHS_1:def 5; then
      xx in (UAp R).(((UAp R).{yy})`) by ROUGHS_2:def 11; then
      not xx in (UAp R).{yy} by B5,XBOOLE_0:def 5;
      hence thesis by a1;
    end; then
    the InternalRel of R is_negative_alliance_in X; then
    R is negative_alliance;
    hence thesis by a1;
  end;
