
theorem
  for S, T being Semilattice st [:S,T:] is meet-continuous holds S is
  meet-continuous & T is meet-continuous
proof
  let S, T be Semilattice such that
A1: [:S,T:] is up-complete and
A2: for x being Element of [:S,T:], D being non empty directed Subset of
  [:S,T:] holds x "/\" sup D = sup ({x} "/\" D);
  hereby
    thus S is up-complete by A1,WAYBEL_2:11;
    set t = the Element of T;
    let s be Element of S, D be non empty directed Subset of S;
    reconsider t9 = {t} as non empty directed Subset of T by WAYBEL_0:5;
    reconsider ST = {[s,t]} as non empty directed Subset of [:S,T:] by
WAYBEL_0:5;
    ex_sup_of [:D,t9:],[:S,T:] by A1,WAYBEL_0:75;
    then
A3: sup [:D,t9:] = [sup proj1 [:D,t9:], sup proj2 [:D,t9:]] by YELLOW_3:46;
    ex_sup_of ST "/\" [:D,t9:],[:S,T:] by A1,WAYBEL_0:75;
    then
A4: sup ({[s,t]} "/\" [:D,t9:]) = [sup proj1 ({[s,t]} "/\" [:D,t9:]), sup
    proj2 ({[s,t]} "/\" [:D,t9:])] by YELLOW_3:46;
    thus sup ({s} "/\" D) = sup (proj1 {[s,t]} "/\" D) by FUNCT_5:12
      .= sup (proj1 {[s,t]} "/\" proj1 [:D,t9:]) by FUNCT_5:9
      .= sup proj1 ({[s,t]} "/\" [:D,t9:]) by Th24
      .= (sup ({[s,t]} "/\" [:D,t9:]))`1 by A4
      .= ([s,t] "/\" sup [:D,t9:] )`1 by A2
      .= [s,t]`1 "/\" (sup [:D,t9:])`1 by Th13
      .= s "/\" (sup [:D,t9:])`1
      .= s "/\" sup proj1 [:D,t9:] by A3
      .= s "/\" sup D by FUNCT_5:9;
  end;
  thus T is up-complete by A1,WAYBEL_2:11;
  set s = the Element of S;
  let t be Element of T, D be non empty directed Subset of T;
  reconsider s9 = {s} as non empty directed Subset of S by WAYBEL_0:5;
  ex_sup_of [:s9,D:],[:S,T:] by A1,WAYBEL_0:75;
  then
A5: sup [:s9,D:] = [sup proj1 [:s9,D:], sup proj2 [:s9,D:]] by YELLOW_3:46;
  reconsider ST = {[s,t]} as non empty directed Subset of [:S,T:] by WAYBEL_0:5
;
  ex_sup_of ST "/\" [:s9,D:],[:S,T:] by A1,WAYBEL_0:75;
  then
A6: sup ({[s,t]} "/\" [:s9,D:]) = [sup proj1 ({[s,t]} "/\" [:s9,D:]), sup
  proj2 ({[s,t]} "/\" [:s9,D:])] by YELLOW_3:46;
  thus sup ({t} "/\" D) = sup (proj2 {[s,t]} "/\" D) by FUNCT_5:12
    .= sup (proj2 {[s,t]} "/\" proj2 [:s9,D:]) by FUNCT_5:9
    .= sup proj2 ({[s,t]} "/\" [:s9,D:]) by Th24
    .= (sup ({[s,t]} "/\" [:s9,D:]))`2 by A6
    .= ([s,t] "/\" sup [:s9,D:] )`2 by A2
    .= [s,t]`2 "/\" (sup [:s9,D:])`2 by Th13
    .= t "/\" (sup [:s9,D:])`2
    .= t "/\" sup proj2 [:s9,D:] by A5
    .= t "/\" sup D by FUNCT_5:9;
end;
