
theorem Th58:
  for L being non empty transitive reflexive RelStr, X,F being Subset of L st
  (for Y being finite Subset of X st Y <> {} holds ex_inf_of Y,L) &
  (for x being Element of L st x in F
  ex Y being finite Subset of X st ex_inf_of Y,L & x = "/\"(Y,L)) &
  (for Y being finite Subset of X st Y <> {} holds "/\"(Y,L) in F)
  holds ex_inf_of X,L iff ex_inf_of F,L
proof
  let L be non empty transitive reflexive RelStr;
  let X,F be Subset of L such that
A1: for Y being finite Subset of X st Y <> {} holds ex_inf_of Y,L and
A2: for x being Element of L st x in F
  ex Y being finite Subset of X st ex_inf_of Y,L & x = "/\"(Y,L) and
A3: for Y being finite Subset of X st Y <> {} holds "/\"(Y,L) in F;
  for x being Element of L holds x is_<=_than X iff x is_<=_than F
  by A1,A2,A3,Th57;
  hence thesis by YELLOW_0:48;
end;
