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 Th26:
  Bottom (a"\/"b) = Bottom a"/\"Bottom b & Bottom (a"/\"b) =
  Bottom a"\/"Bottom b
proof
A1: {Bottom c: c in {a,b}} = {Bottom a,Bottom b}
  proof
    thus {Bottom c: c in {a,b}} c= {Bottom a,Bottom b}
    proof
      let x be object;
      assume x in {Bottom c: c in {a,b}};
      then consider c such that
A2:   x = Bottom c and
A3:   c in {a,b};
      c = a or c = b by A3,TARSKI:def 2;
      hence thesis by A2,TARSKI:def 2;
    end;
    let x be object;
    assume x in {Bottom a,Bottom b};
    then x = Bottom a & a in {a,b} or x = Bottom b & b in {a,b} by TARSKI:def 2
;
    hence thesis;
  end;
  a"\/"b = "\/"{a,b} & Bottom a"/\"Bottom b = "/\"{Bottom a,Bottom b} by
LATTICE3:43;
  hence Bottom (a"\/"b) = Bottom a"/\"Bottom b by A1,Th24;
  Bottom a"\/"Bottom b = "\/"{Bottom a,Bottom b} & a"/\"b = "/\"{a,b} by
LATTICE3:43;
  hence thesis by A1,Th25;
end;
