theorem Th24:
  Bottom "\/"(X,Q) = "/\"({Bottom a: a in X}, Q)
proof
  {"/\"({Bottom a: a in X}, Q) [*] b: b in X} is_less_than Bottom Q
  proof
    let c;
    assume c in {"/\"({Bottom a: a in X}, Q) [*] b: b in X};
    then consider b such that
A1: c = "/\"({Bottom a: a in X}, Q) [*] b & b in X;
    Bottom b in {Bottom a: a in X} by A1;
    then "/\"({Bottom a: a in X}, Q) [= Bottom b by LATTICE3:38;
    hence c [= Bottom Q by A1,Th12;
  end;
  then "\/"({"/\" ({Bottom a: a in X}, Q) [*] b: b in X}, Q) [= Bottom Q by
LATTICE3:def 21;
  then "/\"({Bottom a: a in X}, Q) [*] "\/"(X,Q) [= Bottom Q by Def5;
  then
A2: "/\"({Bottom a: a in X}, Q) [= Bottom "\/"(X,Q) by Th12;
  Bottom "\/"(X,Q) is_less_than {Bottom a: a in X}
  proof
    let b;
    assume b in {Bottom a: a in X};
    then consider a such that
A3: b = Bottom a and
A4: a in X;
    a [= "\/"(X,Q) by A4,LATTICE3:38;
    hence thesis by A3,Th13;
  end;
  then Bottom "\/"(X,Q) [= "/\"({Bottom a: a in X}, Q) by LATTICE3:39;
  hence thesis by A2,LATTICES:8;
end;
