reserve X for BCI-algebra;
reserve x,y,z for Element of X;
reserve i,j,k,l,m,n for Nat;
reserve f,g for sequence of the carrier of X;
reserve B,P for non empty Subset of X;

theorem
  X is weakly-positive-implicative BCI-algebra implies X is BCI-algebra
  of 0,1,1,1
proof
  assume
A1: X is weakly-positive-implicative BCI-algebra;
  for x,y being Element of X holds Polynom (0,1,x,y) = Polynom (1,1,y,x)
  proof
    let x,y be Element of X;
A2: (x\(x\y))\(y\x)=((y\(y\x))\(y\x))\(x\y) by A1,BCIALG_1:85;
    ((x,(x\y)) to_power 1,(y\x)) to_power 1 = (x,(x\y)) to_power 1 \ (y\x)
    by BCIALG_2:2
      .= (x\(x\y))\(y\x) by BCIALG_2:2
      .= (((y\(y\x))\(y\x)),(x\y)) to_power 1 by A2,BCIALG_2:2
      .= ((y,(y\x)) to_power 2,(x\y)) to_power 1 by BCIALG_2:3;
    hence thesis;
  end;
  hence thesis by Def3;
end;
