 reserve U for set,
         X, Y for Subset of U;

theorem Th12:
  for U being non empty set, A1, A2, B1, B2 being Subset of U st
        A1 c= A2 & B1 c= B2 holds
    INTERSECTION (Inter (A1,A2), Inter (B1,B2)) =
      { C where C is Subset of U : A1 /\ B1 c= C & C c= A2 /\ B2 }
  proof
    let U be non empty set,
        A1, A2, B1, B2 be Subset of U;
    assume that
A1: A1 c= A2 and
A2: B1 c= B2;
    set A = Inter (A1,A2), B = Inter (B1,B2);
    set LAB = A1 /\ B1;
    set UAB = A2 /\ B2;
    set IT = INTERSECTION (Inter (A1,A2), Inter (B1,B2));
    thus IT c= { C where C is Subset of U : LAB c= C & C c= UAB }
    proof
      let x be object;
     reconsider xx=x as set by TARSKI:1;
      assume x in IT; then
      consider X, Y be set such that
A3:   X in A & Y in B & x = X /\ Y by SETFAM_1:def 5;
      xx c= X by A3,XBOOLE_1:17; then
A4:   x is Subset of U by A3,XBOOLE_1:1;
A5:   A1 c= X by Th1,A3;
      B1 c= Y by Th1,A3; then
A6:   LAB c= xx by A5,A3,XBOOLE_1:27;
A7:   X c= A2 by Th1,A3;
      Y c= B2 by Th1,A3; then
      xx c= UAB by A7,A3,XBOOLE_1:27;
      hence thesis by A6,A4;
    end;
    let x be object;
     reconsider xx=x as set by TARSKI:1;
    assume x in { C where C is Subset of U : LAB c= C & C c= UAB };
    then consider C9 being Subset of U such that
A8: C9 = x & LAB c= C9 & C9 c= UAB;
    set x1 = (xx \/ A1) /\ A2;
    set x2 = (xx \/ B1) /\ B2;
A9: LAB \/ xx = x by A8,XBOOLE_1:12;
A10: UAB /\ xx = x by A8,XBOOLE_1:28;
A11: x1 /\ x2 = (xx \/ A1) /\ (A2 /\ ((xx \/ B1) /\ B2)) by XBOOLE_1:16
            .= (xx \/ A1) /\ ((xx \/ B1) /\ (B2 /\ A2)) by XBOOLE_1:16
            .= (xx \/ A1) /\ (xx \/ B1) /\ UAB by XBOOLE_1:16
            .= x by A9,A10,XBOOLE_1:24;
 A1 /\ A2 = A1 by A1,XBOOLE_1:28;
    then x1 = (xx /\ A2) \/ A1 by XBOOLE_1:23; then
A12: A1 c= x1 by XBOOLE_1:7;
    x1 c= A2 by XBOOLE_1:17; then
A13: x1 in A by A12;
 B1 /\ B2 = B1 by A2,XBOOLE_1:28;
    then x2 = (xx /\ B2) \/ B1 by XBOOLE_1:23; then
A14: B1 c= x2 by XBOOLE_1:7;
    x2 c= B2 by XBOOLE_1:17; then
    x2 in B by A14;
    hence thesis by A11,A13,SETFAM_1:def 5;
  end;
