
theorem Th19:
  for L being antisymmetric RelStr for a,b,c being Element of L
holds c = a"/\"b & ex_inf_of {a,b},L iff c <= a & c <= b & for d being Element
  of L st d <= a & d <= b holds c >= d
proof
  let L be antisymmetric RelStr;
  let a,b,c be Element of L;
  hereby
    assume that
A1: c = a"/\"b and
A2: ex_inf_of {a,b},L;
    consider c1 being Element of L such that
A3: {a,b} is_>=_than c1 and
A4: for d being Element of L st {a,b} is_>=_than d holds c1 >= d by A2;
A5: now
      let d be Element of L;
      assume a >= d & b >= d;
      then {a,b} is_>=_than d by Th8;
      hence c1 >= d by A4;
    end;
    a >= c1 & b >= c1 by A3,Th8;
    hence
    c <= a & c <= b & for d being Element of L st d <= a & d <= b holds c
    >= d by A1,A5,LATTICE3:def 14;
  end;
  assume that
A6: c <= a & c <= b and
A7: for d being Element of L st d <= a & d <= b holds c >= d;
  thus c = a"/\"b by A6,A7,LATTICE3:def 14;
  now
    take c;
    thus c is_<=_than {a,b} by A6,Th8;
    let d be Element of L;
    assume d is_<=_than {a,b};
    then d <= a & d <= b by Th8;
    hence c >= d by A7;
  end;
  hence thesis by Th16;
end;
