
theorem Th10:
  for L being up-complete Semilattice for D being non empty
  directed Subset of [:L,L:] holds "\/" ({ sup X where X is non empty directed
Subset of L: ex x being Element of L st X = {x} "/\" proj2 D & x in proj1 D},L)
= "\/" (union {X where X is non empty directed Subset of L: ex x being Element
  of L st X = {x} "/\" proj2 D & x in proj1 D},L)
proof
  let L be up-complete Semilattice, D be non empty directed Subset of [:L,L:];
  defpred P[set] means ex x being Element of L st $1 = {x} "/\" proj2 D & x in
  proj1 D;
A1: "\/" ({ sup X where X is non empty directed Subset of L: P[X] },L) <=
  "\/"(union {X where X is non empty directed Subset of L: P[X]},L) by Th9;
A2: union {X where X is non empty directed Subset of L: P[X]} is_<=_than
  "\/" ({ "\/"(X,L) where X is non empty directed Subset of L: P[X] },L)
  proof
    let a be Element of L;
    assume a in union {X where X is non empty directed Subset of L: P[X]};
    then consider b being set such that
A3: a in b and
A4: b in {X where X is non empty directed Subset of L: P[X]} by TARSKI:def 4;
    consider c being non empty directed Subset of L such that
A5: b = c and
A6: P[c] by A4;
    "\/"(c,L) in { "\/" (X,L) where X is non empty directed Subset of L:
    P[X] } by A6;
    then
A7: "\/"(c,L) <= "\/" ({ "\/"(X,L) where X is non empty directed Subset
    of L: P[X] },L) by Th8,YELLOW_4:1;
    ex_sup_of c,L by WAYBEL_0:75;
    then a <= "\/"(c,L) by A3,A5,YELLOW_4:1;
    hence
    a <= "\/" ({ "\/"(X,L) where X is non empty directed Subset of L: P[X
    ] },L) by A7,YELLOW_0:def 2;
  end;
  ex_sup_of union {X where X is non empty directed Subset of L: P[X]},L by Th7;
  then "\/"(union {X where X is non empty directed Subset of L: P[X]},L) <=
  "\/" ({ "\/"(X,L) where X is non empty directed Subset of L: P[X] },L) by A2,
YELLOW_0:def 9;
  hence thesis by A1,ORDERS_2:2;
end;
