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 Th20:
  for X being BCK-algebra of i,j,m,n holds for x,y being Element
  of X holds (x,y) to_power (j+1) = (x,y) to_power (m+1)
proof
  let X be BCK-algebra of i,j,m,n;
  let x,y be Element of X;
A1: (x\y)\x = (x\x)\y by BCIALG_1:7
    .= y` by BCIALG_1:def 5
    .= 0.X by BCIALG_1:def 8;
  then
A2: Polynom (i+1,j,(x\y),x) = ((x\y),(x\(x\y))) to_power j by BCIALG_2:5
    .= ((x\(x\(x\y))),(x\(x\y))) to_power j by BCIALG_1:8
    .= (((x,(x\(x\y))) to_power 1),(x\(x\y))) to_power j by BCIALG_2:2
    .= (x,(x\(x\y))) to_power (j+1) by BCIALG_2:10
    .= (x,y) to_power (j+1) by BCIALG_2:8;
A3: X is BCI-algebra of i+1,j,m,n+1 by Th14;
  Polynom (m,n+1,x,(x\y)) = (x,(x\(x\y))) to_power (m+1) by A1,BCIALG_2:5
    .= (x,y) to_power (m+1) by BCIALG_2:8;
  hence thesis by A2,A3,Def3;
end;
