
theorem Th18:
  for L being antisymmetric RelStr for a,b,c being Element of L
holds c = a"\/"b & ex_sup_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_sup_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 13;
  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 13;
  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 Th15;
end;
