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, p be object
   st A c= B & B /\ C = {p} & p in A holds A /\ C = {p}
proof
  let A,B,C be set, p be object such that
A1: A c= B;
  assume
A2: B /\ C = {p};
  p in B /\ C by A2,TARSKI:def 1;
  then
A3: p in C by XBOOLE_0:def 4;
  assume p in A;
  then p in A /\ C by A3,XBOOLE_0:def 4;
  hence thesis by A2,Lm3,A1,XBOOLE_1:26;
end;
