reserve A, B, X, Y for set;

theorem
  for L1 being meet-continuous Semilattice, L2 being non empty reflexive
  RelStr st the RelStr of L1 = the RelStr of L2 holds L2 is meet-continuous
proof
  let L1 be meet-continuous Semilattice, L2 be non empty reflexive RelStr;
  assume
A1: the RelStr of L1 = the RelStr of L2;
  hence L2 is up-complete by WAYBEL_8:15;
  let x be Element of L2, D be non empty directed Subset of L2;
  reconsider E = D as non empty directed Subset of L1 by A1,WAYBEL_0:3;
  reconsider y = x as Element of L1 by A1;
A2: {x} "/\" D = {y} "/\" E by A1,Th11;
  reconsider yy = {y} as non empty directed Subset of L1 by WAYBEL_0:5;
A3: ex_sup_of yy "/\" E, L1 by WAYBEL_0:75;
  ex_sup_of E, L1 by WAYBEL_0:75;
  then sup D = sup E by A1,YELLOW_0:26;
  hence x "/\" sup D = y "/\" sup E by A1,Th9
    .= sup ({y} "/\" E) by WAYBEL_2:def 6
    .= sup ({x} "/\" D) by A1,A2,A3,YELLOW_0:26;
end;
