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

theorem Prop16L: :: Proposition 16 (L)
  for R1, R2 being non empty RelStr st
    the carrier of R1 = the carrier of R2 &
      the InternalRel of R1 c= the InternalRel of R2 holds
      LAp R2 cc= LAp R1
  proof
    let R1, R2 be non empty RelStr;
    assume
A0: the carrier of R1 = the carrier of R2 &
      the InternalRel of R1 c= the InternalRel of R2;
A5: dom LAp R1 = bool the carrier of R1 by FUNCT_2:def 1;
    for x being set st x in dom LAp R2 holds
      (LAp R2).x c= (LAp R1).x
    proof
      let x be set;
      assume
A2:   x in dom LAp R2;
      then reconsider x1 = x as Subset of R1 by A0;
A3:   (LAp R1).x = LAp x1 by ROUGHS_2:def 10;
      reconsider x2 = x as Subset of R2 by A2;
      LAp x2 c= LAp x1
      proof
        let y be object;
        assume y in LAp x2; then
        y in { x where x is Element of R2 : Class (the
          InternalRel of R2, x) c= x2 } by ROUGHS_1:def 4; then
        consider xx being Element of R2 such that
C1:     xx = y & Class (the InternalRel of R2, xx) c= x2;
        reconsider xxx = xx as Element of R2;
        Class (the InternalRel of R1, xx) c=
          Class (the InternalRel of R2, xx) by RELAT_1:124,A0; then
C2:     Class (the InternalRel of R1, xx) c= x1 by C1,XBOOLE_1:1;
        reconsider xx1 = xx as Element of R1 by A0;
        xx1 in { x where x is Element of R1 :
          Class (the InternalRel of R1, x) c= x1 } by C2;
        hence thesis by C1,ROUGHS_1:def 4;
      end;
      hence thesis by A3,ROUGHS_2:def 10;
    end;
    hence thesis by A5,A0,ALTCAT_2:def 1;
  end;
