
theorem
  for L being with_infima Poset, X being Subset of L
  st ex_inf_of X,L or L is complete holds inf X = inf fininfs X
proof
  let L be with_infima Poset, X be Subset of L;
  assume ex_inf_of X,L or L is complete;
  then
A1: ex_inf_of X,L by YELLOW_0:17;
A2: now
    let Y be finite Subset of X;
    Y c= the carrier of L by XBOOLE_1:1;
    hence Y <> {} implies ex_inf_of Y,L by YELLOW_0:55;
  end;
A3: now
    let x be Element of L;
    assume x in fininfs X;
    then ex Y being finite Subset of X st x = "/\"(Y,L) & ex_inf_of Y,L;
    hence ex Y being finite Subset of X st ex_inf_of Y,L & x = "/\"(Y,L);
  end;
  now
    let Y be finite Subset of X;
    reconsider Z = Y as Subset of L by XBOOLE_1:1;
    assume Y <> {};
    then ex_inf_of Z,L by YELLOW_0:55;
    hence "/\"(Y,L) in fininfs X;
  end;
  hence thesis by A1,A2,A3,Th59;
end;
