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

theorem Prop137H: :: Proposition 13 (7H") for singletons - original proofs
  for R being finite negative_alliance non empty RelStr,
      x being Element of R holds
    (UAp R).(((UAp R).{x})`) c= ((UAp R).{x})`
  proof
    let R be finite negative_alliance non empty RelStr,
        x be Element of R;
    set H = UAp R;
    set L = Flip H;
w1: H.{} = {} by UApEmpty;
w5: for X,Y being Subset of R holds H.(X \/ Y) = H.X \/ H.Y by UApAdditive;
    set RR = GeneratedRelStr H;
w3: UAp R = UAp GeneratedRelStr H by KeyTheorem,w1,w5,ROUGHS_4:def 9;
WZ: the InternalRel of RR = the InternalRel of R by Corr3A,w3;
W1: the InternalRel of R is_negative_alliance_in the carrier of R by DefNA;
    let y be object;
    assume
w2: y in H.((H.{x})`);
    reconsider Hx = (H.{x})` as Subset of R;
    y in UAp Hx by w2,ROUGHS_2:def 11; then
    consider z being object such that
O1: z in Class (the InternalRel of R, y) & z in Hx by XBOOLE_0:3,ROUGHS_2:7;
p1: [y,z] in the InternalRel of R by O1,RELAT_1:169;
    reconsider zz = z, yy = y as Element of RR by O1,w2;
    reconsider xx = x as Element of RR;
    not zz in H.{x} by O1,XBOOLE_0:def 5; then
p2: not [zz,x] in the InternalRel of RR by GRDef;
    set W = the carrier of R, I = the InternalRel of R;
    not [yy,xx] in I by W1,WZ,p2,p1; then
    not yy in H.{xx} by GRDef,WZ;
    hence thesis by XBOOLE_0:def 5;
  end;
