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;
reserve F1,F2 for Filter of I;
reserve a,b,c for Element of B;
reserve o1,o2 for BinOp of F;

theorem Th50:
  for p9,q9 being Element of latt F st p = p9 & q = q9 holds p"\/"
  q = p9"\/"q9 & p"/\"q = p9"/\"q9
proof
  let p9,q9 be Element of latt F such that
A1: p = p9 & q = q9;
  consider o1,o2 such that
A2: o1 = (the L_join of L)||F and
A3: o2 = (the L_meet of L)||F and
A4: latt F = LattStr (#F, o1, o2#) by Def9;
  dom o1 = [:F,F:] by FUNCT_2:def 1;
  then [p,q] in dom o1 by A1,A4,ZFMISC_1:87;
  hence p"\/"q = p9"\/"q9 by A1,A2,A4,FUNCT_1:47;
  dom o2 = [:F,F:] by FUNCT_2:def 1;
  then [p,q] in dom o2 by A1,A4,ZFMISC_1:87;
  hence thesis by A1,A3,A4,FUNCT_1:47;
end;
