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

theorem Prop16H: :: Proposition 16 (H)
  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
      UAp R1 cc= UAp R2
  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;
a1: dom UAp R2 = bool the carrier of R2 by FUNCT_2:def 1;
    for x being set st x in dom UAp R1 holds
      (UAp R1).x c= (UAp R2).x
    proof
      let x be set;
      assume
A2:   x in dom UAp R1;
      then reconsider x1 = x as Subset of R1;
      reconsider x2 = x as Subset of R2 by A0,A2;
A4:   (UAp R2).x = UAp x2 by ROUGHS_2:def 11;
      UAp x1 c= UAp x2
      proof
        let y be object;
        assume y in UAp x1; then
        y in { x where x is Element of R1 : Class (the
          InternalRel of R1, x) meets x1 } by ROUGHS_1:def 5; then
        consider xx being Element of R1 such that
C1:     xx = y & Class (the InternalRel of R1, xx) meets x1;
        reconsider xxx = xx as Element of R2 by A0;
        consider z being object such that
C2:     z in Class (the InternalRel of R1, xx) & z in x1 by C1,XBOOLE_0:3;
        Class (the InternalRel of R1, xx) c=
          Class (the InternalRel of R2, xx) by RELAT_1:124,A0; then
        Class (the InternalRel of R2, xx) meets x2 by C2,XBOOLE_0:3; then
        xxx in { x where x is Element of R2 :
          Class (the InternalRel of R2, x) meets x2 };
        hence thesis by C1,ROUGHS_1:def 5;
      end;
      hence thesis by ROUGHS_2:def 11,A4;
    end;
    hence thesis by a1,A0,ALTCAT_2:def 1;
  end;
