reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;
reserve I,J for Ideal of L,
  F for Filter of L;
reserve D for non empty Subset of L,
  D9 for non empty Subset of L.:;
reserve D1,D2 for non empty Subset of L,
  D19,D29 for non empty Subset of L.:;
reserve B for B_Lattice,
  IB,JB for Ideal of B,
  a,b for Element of B;
reserve a9 for Element of (B qua Lattice).:;
reserve P for non empty ClosedSubset of L,
  o1,o2 for BinOp of P;

theorem Th60:
  (the L_join of L)||P is BinOp of P & (the L_meet of L)||P is BinOp of P
proof
  dom join(L) = [:carr(L),carr(L):] by FUNCT_2:def 1;
  then
A1: dom(join(L)||P) = [:P,P:] by RELAT_1:62;
  rng (join(L)||P) c= P
  proof
    let x be object;
    assume x in rng (join(L)||P);
    then consider y being object such that
A2: y in [:P,P:] and
A3: x = (join(L)||P).y by A1,FUNCT_1:def 3;
    consider p1,p2 being Element of P such that
A4: y = [p1,p2] by A2,DOMAIN_1:1;
    x = p1"\/"p2 by A1,A3,A4,FUNCT_1:47;
    hence thesis by LATTICES:def 25;
  end;
  hence join(L)||P is BinOp of P by A1,FUNCT_2:def 1,RELSET_1:4;
  dom met(L) = [:carr(L),carr(L):] by FUNCT_2:def 1;
  then
A5: dom (met(L)||P) = [:P,P:] by RELAT_1:62;
  rng (met(L)||P) c= P
  proof
    let x be object;
    assume x in rng (met(L)||P);
    then consider y being object such that
A6: y in [:P,P:] and
A7: x = (met(L)||P).y by A5,FUNCT_1:def 3;
    consider p1,p2 being Element of P such that
A8: y = [p1,p2] by A6,DOMAIN_1:1;
    x = p1"/\"p2 by A5,A7,A8,FUNCT_1:47;
    hence thesis by LATTICES:def 24;
  end;
  hence thesis by A5,FUNCT_2:def 1,RELSET_1:4;
end;
