reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem
  (id X)*(id Y) = id(X /\ Y)
proof
A1: dom((id X)*(id Y)) = dom((id X)) /\ Y by Th19
    .= X /\ Y;
A2: z in X /\ Y implies ((id X)*(id Y)).z = (id(X /\ Y)).z
  proof
    assume
A3: z in X /\ Y;
    then
A4: z in X by XBOOLE_0:def 4;
A5: z in Y by A3,XBOOLE_0:def 4;
    thus ((id X)*(id Y)).z = (id X).((id Y).z) by A1,A3,Th12
      .= (id X).z by A5,Th17
      .= z by A4,Th17
      .= (id(X /\ Y)).z by A3,Th17;
  end;
  X /\ Y = dom id(X /\ Y);
  hence thesis by A1,A2,Th2;
end;
