
theorem Th17:

:: See a parenthetical remark in the middle of p. 106.
:: This fact is needed in the proof of II-1.11(ii), p. 105.
  for T being non empty TopSpace, S being irreducible Subset of T,
  V being Element of InclPoset the topology of T st V = S` holds V is prime
proof
  let T be non empty TopSpace, S be irreducible Subset of T, V be Element of
  InclPoset the topology of T such that
A1: V =S`;
  set sL = the topology of T;
  let X, Y be Element of InclPoset sL;
A2: the carrier of InclPoset the topology of T = the topology of T by
YELLOW_1:1;
  then X in sL & Y in sL;
  then reconsider X9 = X, Y9 = Y as Subset of T;
  X9 /\ Y9 in sL by A2,PRE_TOPC:def 1;
  then
A3: X /\ Y = X "/\" Y by YELLOW_1:9;
  assume X "/\" Y <= V;
  then X /\ Y c= V by A3,YELLOW_1:3;
  then S c= (X9/\Y9)` by A1,Lm1;
  then S c= X9` \/ Y9` by XBOOLE_1:54;
  then S = (X9` \/ Y9`)/\S by XBOOLE_1:28;
  then
A4: S = (X9`)/\S \/ (Y9`)/\S by XBOOLE_1:23;
A5: S is closed by YELLOW_8:def 3;
  Y9 is open by A2,PRE_TOPC:def 2;
  then Y9` is closed;
  then
A6: (Y9`)/\S is closed by A5;
  X9 is open by A2,PRE_TOPC:def 2;
  then X9` is closed;
  then (X9`)/\S is closed by A5;
  then S = (X9`)/\S or S = (Y9`)/\S by A6,A4,YELLOW_8:def 3;
  then S c= X9` or S c= Y9` by XBOOLE_1:17;
  then X c= V or Y c= V by A1,Lm1;
  hence X <= V or Y <= V by YELLOW_1:3;
end;
