reserve m,n,s for non zero Element of NAT;

theorem Th1:
  for u1,v1,w1 be Element of n-tuples_on BOOLEAN
  holds Op-XOR(Op-XOR(u1,v1),w1) = Op-XOR(u1,Op-XOR(v1,w1))
proof
  let u1,v1,w1 be Element of n-tuples_on BOOLEAN;
A1: len Op-XOR(Op-XOR(u1,v1),w1) = n by Lm1;
A2: len Op-XOR(u1,Op-XOR(v1,w1)) = n by Lm1;
  now let i be Nat;
    assume 1 <= i & i <= len Op-XOR(Op-XOR(u1,v1),w1);
    then
A3: i in Seg n by A1;
    thus (Op-XOR(Op-XOR(u1,v1),w1)).i
    = (Op-XOR(u1,v1)).i 'xor' w1.i by DESCIP_1:def 4,A3
    .= (u1.i 'xor' v1.i) 'xor' w1.i by DESCIP_1:def 4,A3
    .= u1.i 'xor' (v1.i 'xor' w1.i) by XBOOLEAN:73
    .= u1.i 'xor' (Op-XOR(v1,w1)).i by DESCIP_1:def 4,A3
    .= (Op-XOR(u1,Op-XOR(v1,w1))).i by DESCIP_1:def 4,A3;
  end;
  hence thesis by Lm1,A2,FINSEQ_1:14;
end;
