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

theorem Th95H: :: Proposition 9 (5H")
  for R being non empty transitive RelStr,
      X being Subset of R holds
    UAp (UAp X) c= UAp X
  proof
    let R be non empty transitive RelStr;
    let X be Subset of R;
    let x be object;
    assume x in UAp (UAp X); then
    x in {y where y is Element of R :
      Class (the InternalRel of R,y) meets UAp X} by ROUGHS_1:def 5; then
    consider y being Element of R such that
A1: y = x & Class (the InternalRel of R,y) meets UAp X;
    consider b being object such that
A2: b in Class (the InternalRel of R,y) /\ UAp X
      by XBOOLE_0:7,XBOOLE_0:def 7,A1;
    b in Class (the InternalRel of R,y) by A2,XBOOLE_0:def 4; then
A3: [y,b] in the InternalRel of R by RELAT_1:169;
    b in UAp X by A2,XBOOLE_0:def 4; then
    b in {t where t is Element of R : Class (the InternalRel of R,t) meets X}
      by ROUGHS_1:def 5; then
    consider c being Element of R such that
A4: c = b & Class (the InternalRel of R,c) meets X;
    consider d being object such that
A5: d in Class (the InternalRel of R,c) /\ X by XBOOLE_0:7,A4,XBOOLE_0:def 7;
AA: d in Class (the InternalRel of R,c) & d in X by XBOOLE_0:def 4,A5;
A6: [c,d] in the InternalRel of R & d in X by RELAT_1:169,AA;
    [y,d] in the InternalRel of R & d in X
      by ORDERS_2:def 3,RELAT_2:def 8,A6,A3,A4; then
    d in Im (the InternalRel of R,y) & d in X by RELAT_1:169; then
    d in Class (the InternalRel of R,y) /\ X by XBOOLE_0:def 4; then
    Class (the InternalRel of R,y) meets X by XBOOLE_0:def 7; then
    y in {t where t is Element of R : Class (the InternalRel of R,t) meets X};
    hence thesis by A1,ROUGHS_1:def 5;
  end;
