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 p-Semisimple BCI-algebra implies X is BCI-algebra of n+j,n,m,m+j+ 1
proof
  assume
A1: X is p-Semisimple BCI-algebra;
A2: for x,y being Element of X holds Polynom (n,n,x,y) = y
  proof
    let x,y be Element of X;
    defpred P[Nat] means $1 <= n implies Polynom ($1,$1,x,y) = y;
    now
      let k;
      assume
A3:   k <= n implies Polynom (k,k,x,y) = y;
      set m=k+1;
A4:   Polynom (m,m,x,y) = Polynom (k,k+1,x,y)\(x\y) by Th9
        .= (Polynom (k,k,x,y)\(y\x))\(x\y) by Th10;
      assume m <= n;
      then Polynom (m,m,x,y) = x\(x\y) by A1,A3,A4,BCIALG_1:def 26,NAT_1:13;
      hence Polynom (m,m,x,y) = y by A1,BCIALG_1:def 26;
    end;
    then
A5: for k st P[k] holds P[k+1];
    Polynom (0,0,x,y) = x\(x\y) by Th7
      .= y by A1,BCIALG_1:def 26;
    then
A6: P[0];
    for n holds P[n] from NAT_1:sch 2(A6,A5);
    hence thesis;
  end;
A7: for x,y being Element of X holds Polynom (m,m+1,x,y) = x
  proof
    let x,y be Element of X;
    defpred P[Nat] means
$1 <= m implies Polynom ($1,$1+1,x,y) = x;
    now
      let k;
      assume
A8:   k <= m implies Polynom (k,k+1,x,y) = x;
      set l=k+1;
A9:   Polynom (l,l+1,x,y) = Polynom (k,(k+1)+1,x,y)\(x\y) by Th9
        .= (Polynom (k,k+1,x,y)\(y\x))\(x\y) by Th10;
      assume l <= m;
      then Polynom (l,l+1,x,y) = (x\(x\y))\(y\x) by A8,A9,BCIALG_1:7,NAT_1:13
        .= y\(y\x) by A1,BCIALG_1:def 26;
      hence Polynom (l,l+1,x,y) = x by A1,BCIALG_1:def 26;
    end;
    then
A10: for k st P[k] holds P[k+1];
    Polynom (0,1,x,y) = Polynom (0,0,x,y)\(y\x) by Th10
      .= (x\(x\y))\(y\x) by Th7
      .= y\(y\x) by A1,BCIALG_1:def 26
      .= x by A1,BCIALG_1:def 26;
    then
A11: P[0];
    for m holds P[m] from NAT_1:sch 2(A11,A10);
    hence thesis;
  end;
  for x,y being Element of X holds Polynom (n+j,n,x,y) = Polynom (m,m+j+1 ,y,x)
  proof
    let x,y be Element of X;
    defpred P[Nat] means $1 <= j implies Polynom (n+$1,n,x,y) =
    Polynom (m,m+$1+1,y,x);
    now
      let k;
      assume
A12:  k <= j implies Polynom (n+k,n,x,y) = Polynom (m,m+k+1,y,x);
      set l=k+1;
      assume l <= j;
      then Polynom (n+l,n,x,y) = Polynom (m,m+k+1,y,x)\(x\y) by A12,Th9,
NAT_1:13
        .= Polynom (m,m+k+1+1,y,x) by Th10;
      hence Polynom (n+l,n,x,y) = Polynom (m,m+l+1,y,x);
    end;
    then
A13: for k st P[k] holds P[k+1];
    Polynom (n+0,n,x,y) = y by A2
      .= Polynom (m,m+0+1,y,x) by A7;
    then
A14: P[0];
    for j holds P[j] from NAT_1:sch 2(A14,A13);
    hence thesis;
  end;
  hence thesis by Def3;
end;
