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 n,0,m,m implies X is BCI-commutative BCI-algebra
proof
A1: for x,y being Element of X st x\y=0.X holds (y,(y\x)) to_power (m+1) <= (
  y,(y\x)) to_power 1
  proof
    let x,y be Element of X;
    defpred P[Nat] means
$1 <= m implies (y,(y\x)) to_power ($1+1)
    <= (y,(y\x)) to_power 1;
    assume
A2: x\y=0.X;
    now
      (0.X,y\x) to_power 1 = ((0.X,y) to_power 1)\((0.X,x) to_power 1) by
BCIALG_2:18;
      then 0.X\(y\x) = ((0.X,y) to_power 1)\((0.X,x) to_power 1) by BCIALG_2:2;
      then
A3:   0.X\(y\x) = (0.X\y)\((0.X,x) to_power 1) by BCIALG_2:2;
      let k;
      assume
A4:   k<= m implies (y,(y\x)) to_power (k+1) <= (y,(y\x)) to_power 1;
      ((0.X\y)\(0.X\x))\(x\y) = 0.X by BCIALG_1:1;
      then (0.X\y)\(0.X\x) = 0.X by A2,BCIALG_1:2;
      then 0.X\(y\x) = 0.X by A3,BCIALG_2:2;
      then (y\y)\(y\x) = 0.X by BCIALG_1:def 5;
      then (y\(y\x))\y = 0.X by BCIALG_1:7;
      then y\(y\x) <= y;
      then (y\(y\x),(y\x)) to_power (k+1) <= (y,(y\x)) to_power (k+1) by
BCIALG_2:19;
      then
      ((y,(y\x)) to_power 1,(y\x)) to_power (k+1) <= (y,(y\x)) to_power (
      k+1) by BCIALG_2:2;
      then
A5:   (y,(y\x)) to_power ((k+1)+1) <= (y,(y\x)) to_power (k+1) by BCIALG_2:10;
      set m1=k+1;
      assume m1<=m;
      hence (y,(y\x)) to_power (m1+1) <= (y,(y\x)) to_power 1 by A4,A5,Th1,
NAT_1:13;
    end;
    then
A6: for k st P[k] holds P[k+1];
    (y,(y\x)) to_power (0+1) \ (y,(y\x)) to_power 1 = 0.X by BCIALG_1:def 5;
    then
A7: P[0] by BCIALG_1:def 11;
    for m holds P[m] from NAT_1:sch 2(A7,A6);
    hence thesis;
  end;
  assume
A8: X is BCI-algebra of n,0,m,m;
  for x,y being Element of X st x\y=0.X holds x = y\(y\x)
  proof
    let x,y be Element of X;
    assume
A9: x\y=0.X;
    Polynom (n,0,x,y) = Polynom (m,m,y,x) by A8,Def3;
    then (x,(y\x)) to_power 0 = ((y,(y\x)) to_power (m+1),0.X) to_power m by A9
,BCIALG_2:5;
    then
    (x,(y\x)) to_power 0 = ((y,(y\x)) to_power (m+1),0.X) to_power m \ 0.
    X by BCIALG_1:2;
    then (x,(y\x)) to_power 0 = ((y,(y\x)) to_power (m+1),0.X) to_power (m+1)
    by BCIALG_2:4;
    then (x,(y\x)) to_power 0 = (y,(y\x)) to_power (m+1) by BCIALG_2:5;
    then x = (y,(y\x)) to_power (m+1) by BCIALG_2:1;
    then x <= (y,(y\x)) to_power 1 by A1,A9;
    then
A10: x <= y\(y\x) by BCIALG_2:2;
    (y\(y\x))\x = (y\x)\(y\x) by BCIALG_1:7;
    then (y\(y\x))\x = 0.X by BCIALG_1:def 5;
    then y\(y\x) <= x;
    hence thesis by A10,Th2;
  end;
  hence thesis by BCIALG_3:def 4;
end;
