
theorem
  for H being distributive complete LATTICE for a being Element of H, X
  being finite Subset of H holds inf ({a} "\/" X) = a "\/" inf 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 "\/" inf A = inf({
  a} "\/" A);
A1: P[{}]
  proof
    reconsider A = {} as Subset of H by XBOOLE_1:2;
    take A;
    thus A = {};
    a <= Top H & {a} "\/" {}H = {} by YELLOW_0:45,YELLOW_4:9;
    hence thesis by YELLOW_0:24;
  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 "\/" inf A = inf({a} "\/" A) by A5;
A9: {a} "\/" C = ({a} "\/" A) \/ ({a} "\/" {x1}) by A7,YELLOW_4:16
      .= ({a} "\/" A) \/ {a "\/" x1} by YELLOW_4:19;
A10: ex_inf_of {a} "\/" A,H & ex_inf_of {a "\/" x1},H by YELLOW_0:17;
    ex_inf_of B,H & ex_inf_of {x},H by YELLOW_0:17;
    hence a "\/" inf C = a "\/" ("/\"(B, H) "/\" "/\"({x}, H)) by YELLOW_2:4
      .= inf({a} "\/" A) "/\" (a "\/" "/\"({x}, H)) by A7,A8,WAYBEL_1:5
      .= inf({a} "\/" A) "/\" (a "\/" x1) by YELLOW_0:39
      .= inf({a} "\/" A) "/\" inf{a "\/" x1} by YELLOW_0:39
      .= inf({a} "\/" C) by A10,A9,YELLOW_2:4;
  end;
A11: X is finite;
  P[X] from FINSET_1:sch 2(A11,A1,A2);
  hence thesis;
end;
