reserve X for BCI-algebra;
reserve X1 for non empty Subset of X;
reserve A,I for Ideal of X;
reserve x,y,z for Element of X;
reserve a for Element of A;

theorem
  BCK-part(X) is p-ideal of X
proof
  set A = BCK-part(X);
A1: now
    let x,y,z be Element of X;
    assume that
A2: (x\z)\(y\z) in A and
A3: y in A;
    ex c being Element of X st (x\z)\(y\z) = c & 0.X<=c by A2;
    then ((x\z)\(y\z))`= 0.X;
    then ((x\z)`)\((y\z)`)=0.X by BCIALG_1:9;
    then
A4: (x`\z`)\((y\z)`)=0.X by BCIALG_1:9;
    ex d being Element of X st y=d & 0.X<=d by A3;
    then y` = 0.X;
    then ((x`\z`)\(0.X\z`))\(x`\0.X)=(x`\0.X)` by A4,BCIALG_1:9;
    then ((x`\z`)\(0.X\z`))\(x`\0.X)=x`` by BCIALG_1:2;
    then 0.X = 0.X\(0.X\x) by BCIALG_1:def 3;
    then 0.X\x = 0.X by BCIALG_1:1;
    then 0.X <= x;
    hence x in A;
  end;
  0.X\0.X=0.X by BCIALG_1:def 5;
  then 0.X <= 0.X;
  then 0.X in A;
  hence thesis by A1,Def5;
end;
