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

theorem Th95L: :: Proposition 9 (5L")
  for R being non empty transitive RelStr, X being Subset of R holds
    LAp X c= LAp (LAp X)
  proof
    let R be non empty transitive RelStr;
    let X be Subset of R;
    let x be object;
    assume x in LAp X; then
    x in {u where u is Element of R : Class (the InternalRel of R,u) c= X}
      by ROUGHS_1:def 4; then
    consider y being Element of R such that
A1: y = x & Class (the InternalRel of R,y) c= X;
    Class (the InternalRel of R,y) c= LAp X
    proof
      let t be object;
      assume t in Class (the InternalRel of R,y); then
B0:   [y,t] in (the InternalRel of R) by RELAT_1:169; then
b1:   t in rng (the InternalRel of R) by XTUPLE_0:def 13;
      Class (the InternalRel of R,t) c= X
      proof
        let s be object;
        assume s in Class (the InternalRel of R,t); then
B2:     [t,s] in the InternalRel of R by RELAT_1:169; then
        s in rng (the InternalRel of R) by XTUPLE_0:def 13; then
        [y,s] in the InternalRel of R
          by B0,B2,b1,RELAT_2:def 8,ORDERS_2:def 3; then
        s in Im (the InternalRel of R, y) by RELAT_1:169;
        hence thesis by A1;
      end; then
      t in {u where u is Element of R : Class (the InternalRel of R,u) c= X}
        by b1;
      hence thesis by ROUGHS_1:def 4;
    end; then
    y in {u where u is Element of R : Class (the InternalRel of R,u) c= LAp X};
    hence thesis by ROUGHS_1:def 4,A1;
  end;
