
theorem Th57:
  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)
  for x being Element of L holds x is_<=_than X iff x is_<=_than F
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;
  let x be Element of L;
  thus x is_<=_than X implies x is_<=_than F
  proof
    assume
A4: x is_<=_than X;
    let y be Element of L;
    assume y in F;
    then consider Y being finite Subset of X such that
A5: ex_inf_of Y,L and
A6: y = "/\"(Y,L) by A2;
    x is_<=_than Y by A4;
    hence x <= y by A5,A6,YELLOW_0:def 10;
  end;
  assume
A7: x is_<=_than F;
  let y be Element of L;
  assume y in X;
  then
A8: {y} c= X by ZFMISC_1:31;
  then
A9: inf {y} in F by A3;
  ex_inf_of {y},L by A1,A8;
  then
A10: {y} is_>=_than inf {y} by YELLOW_0:def 10;
A11: inf {y} >= x by A7,A9;
  y >= inf {y} by A10,YELLOW_0:7;
  hence thesis by A11,ORDERS_2:3;
end;
