
theorem Th15:
  for H being distributive complete LATTICE for a being Element of
  H, X being finite Subset of H holds sup ({a} "/\" X) = a "/\" sup X
proof
  let H be distributive complete LATTICE, a be Element of H, X be finite
  Subset of H;
  defpred P[set] means ex A being Subset of H st A = $1 & a "/\" sup A = sup({
  a} "/\" A);
A1: P[{}]
  proof
    reconsider A = {} as Subset of H by XBOOLE_1:2;
    take A;
    thus A = {};
    Bottom H <= a & {a} "/\" {}H = {} by YELLOW_0:44,YELLOW_4:36;
    hence thesis by YELLOW_0:25;
  end;
A2: for x, B being set st x in X & B c= X & P[B] holds P[B \/ {x}]
  proof
    let x, B be set such that
A3: x in X and
A4: B c= X and
A5: P[B];
    reconsider x1 = x as Element of H by A3;
A6: {x1} c= the carrier of H;
    B c= the carrier of H by A4,XBOOLE_1:1;
    then reconsider C = B \/ {x} as Subset of H by A6,XBOOLE_1:8;
    take C;
    thus C = B \/ {x};
    consider A being Subset of H such that
A7: A = B and
A8: a "/\" sup A = sup({a} "/\" A) by A5;
A9: {a} "/\" C = ({a} "/\" A) \/ ({a} "/\" {x1}) by A7,YELLOW_4:43
      .= ({a} "/\" A) \/ {a "/\" x1} by YELLOW_4:46;
A10: ex_sup_of {a} "/\" A,H & ex_sup_of {a "/\" x1},H by YELLOW_0:17;
    ex_sup_of B,H & ex_sup_of {x},H by YELLOW_0:17;
    hence a "/\" sup C = a "/\" ("\/"(B, H) "\/" "\/"({x}, H)) by YELLOW_2:3
      .= sup({a} "/\" A) "\/" (a "/\" "\/"({x}, H)) by A7,A8,WAYBEL_1:def 3
      .= sup({a} "/\" A) "\/" (a "/\" x1) by YELLOW_0:39
      .= sup({a} "/\" A) "\/" sup{a "/\" x1} by YELLOW_0:39
      .= sup({a} "/\" C) by A10,A9,YELLOW_2:3;
  end;
A11: X is finite;
  P[X] from FINSET_1:sch 2(A11,A1,A2);
  hence thesis;
end;
