
theorem Th5:
  for X being set st X is c=directed for a,b being set st a in X &
  b in X ex c being set st a \/ b c= c & c in X
proof
  let X be set;
  assume
A1: for Y being finite Subset of X ex a being set st union Y c= a & a in X;
  let a,b be set;
  assume a in X & b in X;
  then
A2: {a,b} is finite Subset of X by ZFMISC_1:32;
  union {a,b} = a \/ b by ZFMISC_1:75;
  hence thesis by A1,A2;
end;
