
theorem LM930:
  for n be Nat,
  z,z1 be Element of BOOLEAN*
  st z=<*> BOOLEAN & z1 =0*n
  holds Class (EqBL2Nat,z) =Class (EqBL2Nat,z1)
  proof
    let n be Nat,
    z,z1 be Element of BOOLEAN*;
    assume AS: z=<*> BOOLEAN & z1 =0*n;
    then
    P1: len z = 0;
    P2: BL2Nat.z = ExAbsval(z) by Def110
    .=0 by D100,P1;
    P3: BL2Nat.z1 = ExAbsval(z1) by Def110;
    per cases;
    suppose n=0;
      hence Class (EqBL2Nat,z) =Class (EqBL2Nat,z1) by AS;
    end;
    suppose nz: n <>0;
      consider n1 be Nat, y be Tuple of n1, BOOLEAN such that
      C2: y = z1 & ExAbsval(z1) =Absval(y) by Def100;
      n1 = len y by CARD_1:def 7
      .=n by CARD_1:def 7,AS,C2; then
      BL2Nat.z1 = 0 by nz,C2,P3,AS, BINARI_3:6; then
      [z,z1] in EqBL2Nat by P2,Def300;
      hence Class (EqBL2Nat,z) =Class (EqBL2Nat,z1) by EQREL_1:35;
    end;
  end;
