reserve X for set;

theorem Th9:
  for X be non empty set for x,y be Element of InclPoset X st x /\
  y in X holds x "/\" y = x /\ y
proof
  let X be non empty set;
  let x,y be Element of InclPoset X;
  assume x /\ y in X;
  then reconsider z = x /\ y as Element of InclPoset X;
  z c= y by XBOOLE_1:17;
  then
A1: z <= y by Th3;
A2: now
    let c be Element of InclPoset X;
    assume c <= x & c <= y;
    then c c= x & c c= y by Th3;
    then c c= z by XBOOLE_1:19;
    hence c <= z by Th3;
  end;
  z c= x by XBOOLE_1:17;
  then z <= x by Th3;
  hence thesis by A1,A2,LATTICE3:def 14;
end;
