
theorem
  for L being sup-Semilattice, C being non empty Subset of L st for x, y
  being Element of L st x in C & y in C holds x <= y or y <= x for Y being non
  empty finite Subset of C holds "\/"(Y,L) in Y
proof
  let L be sup-Semilattice, C be non empty Subset of L such that
A1: for x, y being Element of L st x in C & y in C holds x <= y or y <= x;
  defpred P[set] means "\/"($1,L) in $1 & ex_sup_of $1,L;
A2: for B1, B2 being non empty Element of Fin C holds P[B1] & P[B2] implies
  P[B1 \/ B2]
  proof
    let B1, B2 be non empty Element of Fin C such that
A3: ( P[B1])& P[B2];
    B1 c= C & B2 c= C by FINSUB_1:def 5;
    then "\/"(B1,L) <= "\/"(B2,L) or "\/"(B2,L) <= "\/"(B1,L) by A1,A3;
    then
A4: "\/"(B1,L) "\/" "\/"(B2,L) = "\/"(B1,L) or "\/"(B1,L) "\/" "\/"(B2,L )
    = "\/"(B2,L) by YELLOW_0:24;
    "\/"(B1 \/ B2,L) = "\/"(B1, L) "\/" "\/"(B2, L) by A3,YELLOW_2:3;
    hence thesis by A3,A4,XBOOLE_0:def 3,YELLOW_2:3;
  end;
  let Y be non empty finite Subset of C;
A5: Y in Fin C by FINSUB_1:def 5;
A6: for x being Element of C holds P[{x}]
  proof
    let x be Element of C;
    "\/"({x},L) = x by YELLOW_0:39;
    hence thesis by TARSKI:def 1,YELLOW_0:38;
  end;
  for B being non empty Element of Fin C holds P[B] from SETWISEO:sch 3(
  A6,A2);
  hence thesis by A5;
end;
