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 Th25:
  Bottom "/\"(X,Q) = "\/"({Bottom a: a in X}, Q)
proof
  X is_greater_than "/\"({Bottom a: a in {Bottom b: b in X}}, Q)
  proof
    let c;
    assume c in X;
    then Bottom c in {Bottom b: b in X};
    then
A1: Bottom Bottom c in {Bottom a: a in {Bottom b: b in X}};
    Bottom Bottom c = c by Th22;
    hence thesis by A1,LATTICE3:38;
  end;
  then
A2: "/\"({Bottom a: a in {Bottom b: b in X}}, Q) [= "/\"(X,Q) by LATTICE3:39;
  {Bottom a: a in {Bottom b: b in X}} c= X
  proof
    let x be object;
    assume x in {Bottom a: a in {Bottom b: b in X}};
    then consider a such that
A3: x = Bottom a & a in {Bottom b: b in X};
    ex b st a = Bottom b & b in X by A3;
    hence thesis by A3,Th22;
  end;
  then "/\"(X,Q) [= "/\"({Bottom a: a in {Bottom b: b in X}}, Q) by LATTICE3:45
;
  then "/\"(X,Q) = "/\"({Bottom a: a in {Bottom b: b in X}}, Q) by A2,
LATTICES:8;
  hence Bottom "/\"(X,Q) = Bottom Bottom "\/"({Bottom a: a in X}, Q) by Th24
    .= "\/"({Bottom a: a in X}, Q) by Th22;
end;
