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 Th19:
  for X,I st I is p-ideal of X holds BCK-part(X) c= I
proof
  let X,I;
  assume
A1: I is p-ideal of X;
  let x be object;
  assume
A2: x in BCK-part(X);
  then
A3: ex x1 being Element of X st x=x1 & 0.X<=x1;
  reconsider x as Element of X by A2;
  0.X\x = 0.X by A3;
  then 0.X\(0.X\x)=0.X by BCIALG_1:2;
  then
A4: (x\x)\(0.X\x)=0.X by BCIALG_1:def 5;
  0.X in I by A1,Def5;
  hence thesis by A1,A4,Def5;
end;
