
theorem Th39:  :: Proposition 6 5H'
  for R being non empty mediate RelStr,
      X being Subset of R holds
    UAp X c= UAp (UAp X)
  proof
    let R be non empty mediate RelStr;
    let X be Subset of R;
    let y be object;
    assume y in UAp X; then
    consider t being Element of R such that
A1: t = y & Class (the InternalRel of R,t) meets X;
    ex w being object st
    w in Class (the InternalRel of R,t) /\ X by A1,XBOOLE_0:def 1; then
    consider w being Element of the carrier of R such that
A2: w in Class (the InternalRel of R,t) /\ X;
A3: w in Class (the InternalRel of R,t) & w in X by XBOOLE_0:def 4,A2; then
    [t,w] in the InternalRel of R by RELAT_1:169; then
    consider z being object such that
A4: z in the carrier of R & [t,z] in the InternalRel of R &
      [z,w] in the InternalRel of R by Def5,Def7;
    reconsider z as Element of R by A4;
A5: z in Class (the InternalRel of R,t) &
      w in Class (the InternalRel of R,z) by A4,RELAT_1:169; then
    Class (the InternalRel of R,z) meets X by A3,XBOOLE_0:def 4; then
A6: z in {x where x is Element of R : Class (the InternalRel of R,x) meets X};
A7: UAp {z} c= UAp (UAp X) by Th15,A6,ZFMISC_1:31;
    z in {z} by TARSKI:def 1; then
    Class (the InternalRel of R,t) meets {z} by A5,XBOOLE_0:def 4; then
    t in {x where x is Element of R :
      Class (the InternalRel of R,x) meets {z} };
    hence thesis by A1,A7;
  end;
