
theorem LM800:
  for A, B be Element of Class EqBL2Nat,
  x,y be Element of BOOLEAN*
  st x in A & y in B
  holds A+B =Class (EqBL2Nat,x+y)
  proof
    let A, B be Element of Class EqBL2Nat,
    x2,y2 be Element of BOOLEAN*;
    assume AS1: x2 in A & y2 in B;
    consider x1, y1 be Element of BOOLEAN* such that
    T1: x1 in A & y1 in B & A+B = Class (EqBL2Nat,x1+y1) by Def500;
    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;
    [x1,x2] in EqBL2Nat by Q1,AS1,T1,EQREL_1:22; then
    R1:BL2Nat.x1= BL2Nat.x2 by Def300;
    [y1,y2] in EqBL2Nat by Q2,AS1,T1,EQREL_1:22; then
    R2:BL2Nat.y1= BL2Nat.y2 by Def300;
    BL2Nat.(x1+y1) =BL2Nat.x1+BL2Nat.y1 by LM240
    .=BL2Nat.(x2+y2) by LM240,R1,R2; then
    [x1+y1,x2+y2] in EqBL2Nat by Def300;
    hence thesis by T1,EQREL_1:35;
  end;
