reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;
reserve H,F for Filter of L;
reserve D for non empty Subset of L;
reserve D1,D2 for non empty Subset of L;
reserve I for I_Lattice,
  i,j,k for Element of I;
reserve B for B_Lattice,
  FB,HB for Filter of B;
reserve I for I_Lattice,
  i,j,k for Element of I,
  DI for non empty Subset of I,
  FI for Filter of I;

theorem
  <.F \/ H.) = { r : ex p,q st p"/\"q [= r & p in F & q in H }
proof
  set X = { r1 : ex p,q st p"/\"q [= r1 & p in F & q in H };
  consider p1 such that
A1: p1 in F by SUBSET_1:4;
  consider q1 such that
A2: q1 in H by SUBSET_1:4;
  p1"/\"q1 in X by A1,A2;
  then reconsider D = X as non empty set;
  D c= the carrier of L
  proof
    let x be object;
    assume x in D;
    then ex r1 st x = r1 & ex p,q st p"/\"q [= r1 & p in F & q in H;
    hence thesis;
  end;
  then reconsider D as non empty Subset of L;
A3: for p,q st p in D & p [= q holds q in D
  proof
    let p,q;
    assume p in D;
    then ex r1 st p = r1 & ex p,q st p"/\"q [= r1 & p in F & q in H;
    then consider p1,q1 such that
A4: p1"/\"q1 [= p and
A5: p1 in F & q1 in H;
    assume p [= q;
    then p1"/\"q1 [= q by A4,LATTICES:7;
    hence thesis by A5;
  end;
  for p,q st p in D & q in D holds p"/\"q in D
  proof
    let p,q;
    assume p in D;
    then
    ex r1 be Element of L st p = r1 & ex p,q st p"/\"q [= r1 & p in F & q in H;
    then consider p1,q1 be Element of L such that
A6: p1"/\"q1 [= p and
A7: p1 in F & q1 in H;
    assume q in D;
    then
    ex r2 be Element of L st q = r2 & ex p,q st p"/\"q [= r2 & p in F & q in H;
    then consider p2,q2 be Element of L such that
A8: p2"/\"q2 [= q and
A9: p2 in F & q2 in H;
A10: p"/\"(p2"/\"q2) [= p "/\" q by A8,LATTICES:9;
    p1"/\"q1"/\"(p2"/\"q2) [= p"/\"(p2"/\"q2) by A6,LATTICES:9;
    then
A11: p1"/\"q1"/\"(p2"/\"q2) [= p"/\"q by A10,LATTICES:7;
A12: p1"/\"q1"/\"(p2"/\"q2) = p1"/\"q1"/\"p2"/\"q2 by LATTICES:def 7
      .= p1"/\"p2"/\"q1"/\"q2 by LATTICES:def 7
      .= p1"/\"p2"/\"(q1"/\"q2) by LATTICES:def 7;
    p1"/\"p2 in F & q1"/\"q2 in H by A7,A9,Th8;
    hence thesis by A12,A11;
  end;
  then reconsider D as Filter of L by A3,Th9;
A13: H c= D
  proof
    let x be object;
    assume x in H;
    then reconsider q = x as Element of H;
    q"/\"p1 [= q by LATTICES:6;
    hence thesis by A1;
  end;
  F c= D
  proof
    let x be object;
    assume x in F;
    then reconsider p = x as Element of F;
    p"/\"q1 [= p by LATTICES:6;
    hence thesis by A2;
  end;
  then F \/ H c= D by A13,XBOOLE_1:8;
  hence <.F \/ H.) c= X by Def4;
  let x be object;
  assume x in X;
  then consider r such that
A14: x = r and
A15: ex p,q st p"/\"q [= r & p in F & q in H;
A16: F \/ H c= <.F \/ H.) by Def4;
  H c= F \/ H by XBOOLE_1:7;
  then
A17: H c= <.F \/ H.) by A16;
  consider p,q such that
A18: p"/\"q [= r and
A19: p in F & q in H by A15;
  F c= F \/ H by XBOOLE_1:7;
  then F c= <.F \/ H.) by A16;
  then p"/\"q in <.F \/ H.) by A19,A17,Th8;
  hence thesis by A14,A18,Th9;
end;
