reserve x,x1,x2,x3,x4,y,y1,y2,y3,y4,z,z1,z2,z2,z4 for object;
reserve X,X1,X2,X3,X4,Y for set;

theorem
  proj2(X /\ Y) c= proj2 X /\ proj2 Y
proof
  let y be object;
  assume y in proj2(X /\ Y);
  then consider x such that
A1: [x,y] in X /\ Y by Def13;
  [x,y] in Y by A1,XBOOLE_0:def 4;
  then
A2: y in proj2 Y by Def13;
  [x,y] in X by A1,XBOOLE_0:def 4;
  then y in proj2 X by Def13;
  hence thesis by A2,XBOOLE_0:def 4;
end;
