reserve w, w1, w2 for Element of ExtREAL;
reserve c, c1, c2 for Complex;
reserve A, B, C, D for complex-membered set;
reserve F, G, H, I for ext-real-membered set;
reserve a, b, s, t, z for Complex;
reserve f, g, h, i, j for ExtReal;
reserve r for Real;
reserve e for set;

theorem Th149:
  a ++ (A /\ B) = (a++A) /\ (a++B)
proof
A1: (a++A) /\ (a++B) c= a ++ (A /\ B)
  proof
    let z;
    assume
A2: z in (a++A) /\ (a++B);
    then z in a++A by XBOOLE_0:def 4;
    then consider c such that
A3: z = a+c and
A4: c in A by Th143;
    z in a++B by A2,XBOOLE_0:def 4;
    then ex c1 st z = a+c1 & c1 in B by Th143;
    then c in A/\B by A3,A4,XBOOLE_0:def 4;
    hence thesis by A3,Th141;
  end;
  a ++ (A /\ B) c= (a++A) /\ (a++B) by Th49;
  hence thesis by A1;
end;
