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

theorem Prop11: :: Proposition 11
  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.X)` c= U.((U.X)`)) &
    (for X,Y being Subset of A holds U.(X \/ Y) = U.X \/ U.Y) holds
  ex R being positive_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.X)` c= U.((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 & not [x,y] in I holds
        ex z being object st z in X & [x,z] in I & not [z,y] in I
    proof
      let x,y be object;
      assume
W1:   x in X & y in X & not [x,y] in I; then
      reconsider xx = x, yy = y as Element of R;
a3:   ((UAp R).{yy})` c= U.(((UAp R).{yy})`) by a0,a1;
      not xx in (UAp R).{yy} by W1,a1; then
      xx in ((UAp R).{yy})` by XBOOLE_0:def 5; then
      xx in (UAp R).(((UAp R).{yy})`) by a3,a1; then
      xx in UAp (((UAp R).{yy})`) by ROUGHS_2:def 11; then
      consider z being object such that
J1:   z in Class (the InternalRel of R, xx) & z in (((UAp R).{yy})`)
        by XBOOLE_0:3,ROUGHS_2:7;
      reconsider zz = z as Element of R by J1;
J2:   [xx,zz] in I by J1,RELAT_1:169;
      not zz in ((UAp R).{yy}) by J1,XBOOLE_0:def 5; then
      not [z,y] in I by a1;
      hence thesis by J2;
    end; then
    the InternalRel of R is_positive_alliance_in X; then
    R is positive_alliance;
    hence thesis by a1;
  end;
