
theorem
  for x,y,z be Element of BOOLEAN*
  st x in rng Nat2BL & y in rng Nat2BL & z in rng Nat2BL
  holds (x+y)+z =x+(y+z)
  proof
    let x,y,z be Element of BOOLEAN*;
    assume AS:x in rng Nat2BL & y in rng Nat2BL & z in rng Nat2BL;
    then
    consider n be Element of NAT
    such that
    A1: x =Nat2BL.n by FUNCT_2:113;
    consider m be Element of NAT such that
    A2: y =Nat2BL.m by AS,FUNCT_2:113;
    consider k be Element of NAT such that
    A3: z =Nat2BL.k by AS,FUNCT_2:113;
    reconsider nm = n+m as Element of NAT by ORDINAL1:def 12;
    reconsider mk = m+k as Element of NAT by ORDINAL1:def 12;
    thus (x+y)+z= Nat2BL.(nm)+ Nat2BL.k by LM100,A1,A2,A3
    .= Nat2BL.((n+m) + k) by LM100
    .= Nat2BL.(n+(mk))
    .= Nat2BL.n + Nat2BL.(mk) by LM100
    .=x+(y+z) by A1,A2,A3,LM100;
  end;
