reserve x,y,z for set;
reserve Q for left-distributive right-distributive complete Lattice-like non
  empty QuantaleStr,
  a, b, c, d for Element of Q;
reserve Q for Quantale,
  a,a9,b,b9,c,d,d1,d2,D for Element of Q;
reserve Q for Girard-Quantale,
  a,a1,a2,b,b1,b2,c,d for Element of Q,
  X for set;

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;
