
theorem Th50:
  for L being non empty reflexive antisymmetric RelStr for X being Subset of L
  holds X c= finsups X & X c= fininfs X
proof
  let L be non empty reflexive antisymmetric RelStr;
  let X be Subset of L;
  hereby
    let x be object;
    assume
A1: x in X;
    then reconsider y = x as Element of L;
    reconsider Y = {x} as finite Subset of X by A1,ZFMISC_1:31;
A2: y = "\/"(Y,L) by YELLOW_0:39;
    ex_sup_of {y},L by YELLOW_0:38;
    hence x in finsups X by A2;
  end;
    let x be object;
    assume
A3: x in X;
    then reconsider y = x as Element of L;
    reconsider Y = {x} as finite Subset of X by A3,ZFMISC_1:31;
A4: y = "/\"(Y,L) by YELLOW_0:39;
    ex_inf_of {y},L by YELLOW_0:38;
    hence x in fininfs X by A4;
end;
