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

theorem Prop13H: :: Proposition 13 (7H")
  for R being finite negative_alliance non empty RelStr,
      X being Subset of R holds
    UAp ((UAp X)`) c= (UAp X)`
  proof
    let R be finite negative_alliance non empty RelStr;
    let X be Subset of R;
    defpred P[Subset of R] means
      UAp ((UAp $1)`) c= (UAp $1)`;
    {}R = {}; then
A1: P[{}the carrier of R];
A2: for B being Subset of R,
        b being Element of R holds
    P[B] & not b in B implies P[B \/ {b}]
    proof
      let B be Subset of R,
          b be Element of R;
      assume
z1:   P[B] & not b in B;
      reconsider Bb = B \/ {b} as Subset of R;
      UAp ((UAp Bb)`) = UAp ((UAp B \/ UAp {b})`) by ROUGHS_2:13
         .= UAp ((UAp B)` /\ (UAp {b})`) by XBOOLE_1:53; then
Z2:   UAp ((UAp Bb)`) c= UAp ((UAp B)`) /\ UAp ((UAp {b})`) by UApCon;
      UAp ((UAp B)`) /\ UAp ((UAp {b})`) c= (UAp B)` /\ UAp ((UAp {b})`)
        by z1,XBOOLE_1:26; then
Z3:   UAp ((UAp Bb)`) c= (UAp B)` /\ UAp ((UAp {b})`) by Z2,XBOOLE_1:1;
      (UAp R).(((UAp R).{b})`) c= ((UAp R).{b})` by Prop137H; then
      (UAp R).(((UAp R).{b})`) c= (UAp {b})` by ROUGHS_2:def 11; then
      (UAp R).((UAp {b})`) c= (UAp {b})` by ROUGHS_2:def 11; then
      UAp ((UAp {b})`) c= (UAp {b})` by ROUGHS_2:def 11; then
z4:   (UAp B)` /\ UAp ((UAp {b})`) c= (UAp B)` /\ ((UAp {b})`) by XBOOLE_1:26;
      (UAp B)` /\ ((UAp {b})`) = (UAp B \/ UAp {b})` by XBOOLE_1:53
        .= (UAp Bb)` by ROUGHS_2:13;
      hence thesis by z4,Z3,XBOOLE_1:1;
    end;
    for B being Subset of R holds P[B] from FinSubIndA1 (A1,A2);
    hence thesis;
  end;
