
theorem Th46:
  for L being up-complete Semilattice st for x being Element of L,
E being non empty directed Subset of L st x <= sup E holds x <= sup ({x} "/\" E
  ) holds inf_op L is directed-sups-preserving
proof
  let L be up-complete Semilattice such that
A1: for x being Element of L, E being non empty directed Subset of L st
  x <= sup E holds x <= sup ({x} "/\" E);
  let D be Subset of [:L,L:];
  assume D is non empty directed;
  then reconsider DD = D as non empty directed Subset of [:L,L:];
  thus inf_op L preserves_sup_of D
  proof
    reconsider D1 = proj1 DD, D2 = proj2 DD as non empty directed Subset of L
    by YELLOW_3:21,22;
    reconsider C = the carrier of L as non empty set;
    set f = inf_op L;
    assume ex_sup_of D,[:L,L:];
    set d2 = sup D2;
    defpred P[set] means ex x being Element of L st $1 = {x} "/\" D2 & x in D1;
    f.:DD is directed by YELLOW_2:15;
    hence ex_sup_of f.:D,L by WAYBEL_0:75;
    {x "/\" y where x, y is Element of L : x in D1 & y in {d2}} c= C
    proof
      let q be object;
      assume q in {x "/\" y where x, y is Element of L : x in D1 & y in {d2}};
      then ex x, y being Element of L st q = x "/\" y & x in D1 & y in {d2};
      hence thesis;
    end;
    then reconsider
    F = {x "/\" y where x, y is Element of L : x in D1 & y in {d2}}
    as Subset of L;
A2: "\/" ({ 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 Th10;
A3: F = { sup ({z} "/\" D2) where z is Element of L : z in D1 }
    proof
      thus F c= { sup ({z} "/\" D2) where z is Element of L : z in D1 }
      proof
        let q be object;
        assume q in F;
        then consider x, y being Element of L such that
A4:     q = x "/\" y and
A5:     x in D1 and
A6:     y in {d2};
        q = x "/\" d2 by A4,A6,TARSKI:def 1
          .= sup ({x} "/\" D2) by A1,Th28;
        hence thesis by A5;
      end;
      let q be object;
A7:   d2 in {d2} by TARSKI:def 1;
      assume q in { sup ({z} "/\" D2) where z is Element of L : z in D1 };
      then consider z being Element of L such that
A8:   q = sup ({z} "/\" D2) and
A9:   z in D1;
      q = z "/\" d2 by A1,A8,Th28;
      hence thesis by A9,A7;
    end;
    thus sup (f.:D) = sup (D1 "/\" D2) by Th33
      .= "\/" ({ "\/" (X,L) where X is non empty directed Subset of L: P[X]
    },L) by A2,Th6
      .= "\/" ({ sup ({x} "/\" D2) where x is Element of L : x in D1 },L) by
Th5
      .= sup ({d2} "/\" D1) by A3,YELLOW_4:def 4
      .= sup D1 "/\" d2 by A1,Th28
      .= f.(sup D1,sup D2) by Def4
      .= f.sup D by Th12;
  end;
end;
