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 BCI-algebra of 0,1,0,0 implies X is BCI-commutative BCI-algebra
proof
  assume
A1: X is BCI-algebra of 0,1,0,0;
  for x,y being Element of X st y\x=0.X holds y = x\(x\y)
  proof
    let x,y be Element of X;
    Polynom (0,1,x,y) = Polynom (0,0,y,x) by A1,Def3;
    then
    (x,(x\y)) to_power 1 \ (y\x) = ((y,(y\x)) to_power 1,(x\y)) to_power 0
    by BCIALG_2:2;
    then (x\(x\y)) \ (y\x) = ((y,(y\x)) to_power 1,(x\y)) to_power 0 by
BCIALG_2:2;
    then
A2: (x\(x\y)) \ (y\x) = (y,(y\x)) to_power 1 by BCIALG_2:1;
    assume y\x=0.X;
    then (x\(x\y)) \ 0.X = y\0.X by A2,BCIALG_2:2;
    then y = (x\(x\y)) \ 0.X by BCIALG_1:2;
    hence thesis by BCIALG_1:2;
  end;
  then for x,y being Element of X st x\y=0.X holds x = y\(y\x);
  hence thesis by BCIALG_3:def 4;
end;
