
theorem Th27:
  for L be transitive antisymmetric with_suprema RelStr for p,q,u
be Element of L st p < q & (for s be Element of L st p < s holds q <= s) & not
  u <= p holds p "\/" u = q "\/" u
proof
  let L be transitive antisymmetric with_suprema RelStr;
  let p,q,u be Element of L;
  assume that
A1: p < q and
A2: for s be Element of L st p < s holds q <= s and
A3: not u <= p;
A4: q "\/" u >= q by YELLOW_0:22;
A5: now
    let v be Element of L;
    assume that
A6: v >= p and
A7: v >= u;
    v > p by A3,A6,A7,ORDERS_2:def 6;
    then v >= q by A2;
    hence q "\/" u <= v by A7,YELLOW_0:22;
  end;
  p <= q by A1,ORDERS_2:def 6;
  then
A8: q "\/" u >= p by A4,ORDERS_2:3;
  q "\/" u >= u by YELLOW_0:22;
  hence thesis by A8,A5,YELLOW_0:22;
end;
