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;

theorem Th10:
  for L being complete Lattice, j being UnOp of L st j is monotone
holds j is \/-distributive iff for X being Subset of L holds j."\/"X = "\/"({j.
  a where a is Element of L: a in X}, L)
proof
  let L be complete Lattice, j be UnOp of L such that
A1: j is monotone;
  thus j is \/-distributive implies for X being Subset of L holds j."\/"X =
  "\/"({j.a where a is Element of L: a in X}, L)
  proof
    assume
A2: for X being Subset of L holds j."\/"X [= "\/" ({j.a where a is
    Element of L: a in X}, L);
    let X be Subset of L;
    {j.a where a is Element of L: a in X} is_less_than j. "\/" X
    proof
      let x be Element of L;
      assume x in {j.a where a is Element of L: a in X};
      then consider a being Element of L such that
A3:   x = j.a and
A4:   a in X;
      a [= "\/"X by A4,LATTICE3:38;
      hence thesis by A1,A3;
    end;
    then
A5: "\/"({j.a where a is Element of L: a in X}, L) [= j. "\/"X by
LATTICE3:def 21;
    j."\/"X [= "\/" ({j.a where a is Element of L: a in X}, L) by A2;
    hence thesis by A5,LATTICES:8;
  end;
  assume
A6: for X being Subset of L holds j."\/"X = "\/"({j.a where a is
  Element of L: a in X}, L);
  let X be Subset of L;
  j."\/"X [= j."\/"X;
  hence thesis by A6;
end;
