reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;
reserve e for object, X,X1,X2,Y1,Y2 for set;

theorem
  for A,B,C being set st A /\ B c= {p} & p in C & C misses B holds A
  \/ C misses B
proof
  let A,B,C be set such that
A1: A /\ B c= {p} and
A2: p in C and
A3: C misses B;
  {p} c= C by A2,Lm1;
  then A /\ B c= C by A1;
  then A /\ B /\ B c= C /\ B by XBOOLE_1:27;
  then
A4: A /\ (B /\ B) c= C /\ B by XBOOLE_1:16;
  (A \/ C) /\ B = A /\ B \/ C /\ B by XBOOLE_1:23
    .= {} by A3,A4,XBOOLE_1:12;
  hence thesis;
end;
