
theorem Th41:
  for L being with_suprema Poset, a,b being Element of L holds sup
  {a,b} = a"\/" b
proof
  let L be with_suprema Poset, a,b be Element of L;
A1: now
    let c be Element of L;
    assume c is_>=_than {a,b};
    then c >= a & c >= b by Th8;
    hence c >= a"\/"b by Th22;
  end;
  a"\/"b >= a & a"\/"b >= b by Th22;
  then
A2: a"\/"b is_>=_than {a,b} by Th8;
  then ex_sup_of {a,b}, L by A1,Th15;
  hence thesis by A2,A1,Def9;
end;
