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

theorem :: Proposition 10 (7H') for singletons
  for R being finite positive_alliance non empty RelStr,
      x being Element of R holds
    ((UAp R).{x})` c= (UAp R).(((UAp R).{x})`)
    proof
      let R be finite positive_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;
WW:   for x,y being Element of RR holds
        [x,y] in the InternalRel of RR iff x in H.{y} by GRDef;
WZ:   the InternalRel of RR = the InternalRel of R by Corr3A,w3;
W1:   the InternalRel of R is_positive_alliance_in the carrier of R
        by DefPA;
      let y be object;
      assume
w2:   y in (H.{x})`; then
w1:   not y in H.{x} by XBOOLE_0:def 5;
      reconsider xx = x, yy = y as Element of RR by w2;
      not [yy,xx] in the InternalRel of RR by w1,GRDef; then
      consider z being object such that
W2:   z in the carrier of R & [y,z] in the InternalRel of R and
W3:   not [z,x] in the InternalRel of RR by W1,WZ;
      reconsider zz = z as Element of RR by W2;
W5:   yy in H.{zz} by W2,GRDef,WZ;
j1:   {z} c= the carrier of R by ZFMISC_1:31,W2;
w6:   z in (H.{x})` by W3,WW,SUBSET_1:29,W2;
      for X, Y being Subset of R holds
        H.(X \/ Y) = H.X \/ H.Y by UApAdditive; then
      H is \/-preserving by ROUGHS_4:def 9; then
      H.{z} c= H.((H.{x})`) by w6,j1,ROUGHS_4:def 8,ZFMISC_1:31;
      hence thesis by W5;
    end;
