
theorem
  for A, B be Element of Class EqBL2Nat
  holds QuBL2Nat.(A+B) =QuBL2Nat.A+QuBL2Nat.B
  proof
    let A, B be Element of Class EqBL2Nat;
    P0: A in Class EqBL2Nat & B in Class EqBL2Nat;
    consider x being object such that
    Q1: x in BOOLEAN* & A = Class (EqBL2Nat,x) by P0,EQREL_1:def 3;
    consider y being object such that
    Q2: y in BOOLEAN* & B = Class (EqBL2Nat,y) by P0,EQREL_1:def 3;
    reconsider x,y as Element of BOOLEAN* by Q1,Q2;
    x in A & y in B by Q1,Q2,EQREL_1:20; then
    A+B = Class (EqBL2Nat,x+y ) by LM800; then
    R2: QuBL2Nat.(A+B) = BL2Nat.(x+y) by LM700
    .=BL2Nat.x+BL2Nat.y by LM240;
    BL2Nat.x = QuBL2Nat.A by Q1,LM700;
    hence thesis by R2,Q2,LM700;
  end;
