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;

theorem Th36:
  X is BCI-algebra of i,j,j+k,i+k implies X is BCK-algebra
proof
  assume
A1: X is BCI-algebra of i,j,j+k,i+k;
  for y be Element of X holds 0.X\y = 0.X
  proof
    let y be Element of X;
A2: (((y,y) to_power ((j+k)+1)),(0.X\y)) to_power (i+k) = (((y,y) to_power
    (j+k)\y),(0.X\y)) to_power (i+k) by BCIALG_2:4
      .= ((y\y,y) to_power (j+k),(0.X\y)) to_power (i+k) by BCIALG_2:7
      .= ((0.X,y) to_power (j+k),(0.X\y)) to_power (i+k) by BCIALG_1:def 5
      .= ((0.X,(0.X\y)) to_power (i+k),y) to_power (j+k) by BCIALG_2:11
      .= (((0.X,0.X) to_power (i+k))\((0.X,y) to_power (i+k)),y) to_power (j
    +k) by BCIALG_2:18
      .= ((0.X\((0.X,y) to_power (i+k))),y) to_power (j+k) by BCIALG_2:6
      .= ((0.X,y) to_power (j+k))\((0.X,y) to_power (i+k)) by BCIALG_2:7
      .= ((0.X,y) to_power j,y) to_power k \ ((0.X,y) to_power (i+k)) by
BCIALG_2:10
      .= ((0.X,y) to_power j,y) to_power k \ ((0.X,y) to_power i,y) to_power
    k by BCIALG_2:10;
A3: ((0.X,y) to_power j,y) to_power k \ ((0.X,y) to_power i,y) to_power k
    <= (0.X,y) to_power j \ (0.X,y) to_power i by BCIALG_2:21;
    Polynom (i,j,0.X,y) = Polynom (j+k,i+k,y,0.X) by A1,Def3;
    then
    (((0.X,(0.X\y)) to_power (i+1)),y) to_power j = (((y,(y\0.X)) to_power
    ((j+k)+1)),(0.X\y)) to_power (i+k) by BCIALG_1:2;
    then
    (((0.X,(0.X\y)) to_power (i+1)),y) to_power j = (((y,y) to_power ((j+k
    )+1)),(0.X\y)) to_power (i+k) by BCIALG_1:2;
    then
A4: ((0.X,y) to_power j,(0.X\y)) to_power (i+1) = (((y,y) to_power ((j+k)+1
    )),(0.X\y)) to_power (i+k) by BCIALG_2:11;
    ((0.X,y) to_power j,(0.X\y)) to_power (i+1) = ((0.X,(0.X\y)) to_power (
    i+1),y) to_power j by BCIALG_2:11
      .= (((0.X,0.X) to_power (i+1))\((0.X,y) to_power (i+1)),y) to_power j
    by BCIALG_2:18
      .= ((0.X\((0.X,y) to_power (i+1))),y) to_power j by BCIALG_2:6
      .= ((0.X,y) to_power j)\((0.X,y) to_power (i+1)) by BCIALG_2:7;
    then
    ((0.X,y) to_power j \ (0.X,y) to_power (i+1)) \ ((0.X,y) to_power j \
    (0.X,y) to_power i) = 0.X by A4,A2,A3;
    then 0.X \ ((0.X,y) to_power i \ (0.X,y) to_power (i+1)) = 0.X by
BCIALG_1:11;
    then
A5: 0.X <= ((0.X,y) to_power i \ (0.X,y) to_power (i+1));
    ((0.X,y) to_power i \ (0.X,y) to_power (i+1)) = (0.X,y) to_power i \
    ((0.X,y) to_power i \ y) by BCIALG_2:4
      .= (0.X,y) to_power i \ (0.X\y,y) to_power i by BCIALG_2:7;
    then (0.X,y) to_power i \ (0.X,y) to_power (i+1) <= 0.X \ (0.X\y) by
BCIALG_2:21;
    then 0.X <= 0.X \ (0.X\y) by A5,Th1;
    then 0.X \ (0.X \ (0.X\y)) = 0.X;
    hence thesis by BCIALG_1:8;
  end;
  then for x being Element of X holds x`=0.X;
  hence thesis by BCIALG_1:def 8;
end;
