reserve a for set;

theorem Th11:
  for L being lower-bounded sup-Semilattice
  for X being non empty Subset of InclPoset Aux L holds
  meet X is auxiliary Relation of L
proof
  let L be with_suprema lower-bounded Poset;
  let X be non empty Subset of InclPoset Aux L;
  X c= the carrier of InclPoset Aux L;
  then
A1: X c= Aux L by YELLOW_1:1;
  set a = the Element of X;
  a in X;
  then a is auxiliary Relation of L by A1,Def8;
  then reconsider ab = meet X as Relation of L by SETFAM_1:7;
A2: ab is auxiliary(i)
  proof
    let x, y be Element of L;
    assume
A3: [x,y] in ab;
    set V = the Element of X;
A4: V in X;
A5: [x,y] in V by A3,SETFAM_1:def 1;
    V is auxiliary Relation of L by A1,A4,Def8;
    hence x <= y by A5,Def3;
  end;
A6: ab is auxiliary(ii)
  proof
    let x, y, z, u be Element of L;
    assume that
A7: u <= x and
A8: [x,y] in ab and
A9: y <= z;
    for Y be set st Y in X holds [u,z] in Y
    proof
      let Y be set;
      assume
A10:  Y in X;
      then
A11:  Y is auxiliary Relation of L by A1,Def8;
      [x,y] in Y by A8,A10,SETFAM_1:def 1;
      hence thesis by A7,A9,A11,Def4;
    end;
    hence thesis by SETFAM_1:def 1;
  end;
A12: ab is auxiliary(iii)
  proof
    let x, y, z be Element of L;
    assume that
A13: [x,z] in ab and
A14: [y,z] in ab;
    for Y be set st Y in X holds [(x "\/" y),z] in Y
    proof
      let Y be set;
      assume
A15:  Y in X;
      then
A16:  Y is auxiliary Relation of L by A1,Def8;
A17:  [x,z] in Y by A13,A15,SETFAM_1:def 1;
      [y,z] in Y by A14,A15,SETFAM_1:def 1;
      hence thesis by A16,A17,Def5;
    end;
    hence thesis by SETFAM_1:def 1;
  end;
  ab is auxiliary(iv)
  proof
    let x be Element of L;
    for Y be set st Y in X holds [Bottom L,x] in Y
    proof
      let Y be set;
      assume Y in X;
      then Y is auxiliary Relation of L by A1,Def8;
      hence thesis by Def6;
    end;
    hence thesis by SETFAM_1:def 1;
  end;
  hence thesis by A2,A6,A12;
end;
