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;
reserve X for BCK-algebra;
reserve X for BCI-algebra;
reserve X for BCK-algebra;
reserve I for Ideal of X;
reserve I for Ideal of X;

theorem
  (for I being Ideal of X holds I is commutative Ideal of X) implies (
  for x,y being Element of X holds (x\y=x iff y\(y\x)=0.X)) & (for x,y being
  Element of X st x\y=x holds y\x=y) & (for x,y,a being Element of X st y <= a
holds (a\x)\(a\y) = y\x) &(for x,y being Element of X holds x\(y\(y\x))=x\y & (
x\y)\((x\y)\x)=x\y) & for x,y,a being Element of X st x <= a holds (a\y)\((a\y)
  \(a\x)) = (a\y)\(x\y)
proof
  assume for I being Ideal of X holds I is commutative Ideal of X;
  then
A1: X is commutative BCK-algebra by Th37;
  hence for x,y being Element of X holds (x\y=x iff y\(y\x)=0.X) by BCIALG_3:9;
  thus for x,y being Element of X st x\y=x holds y\x=y by A1,BCIALG_3:7;
  thus for x,y,a being Element of X st y <= a holds (a\x)\(a\y) = y\x by A1,
BCIALG_3:8;
  thus for x,y being Element of X holds x\(y\(y\x))=x\y &(x\y)\((x\y)\x)=x\y
  by A1,BCIALG_3:10;
  thus thesis by A1,BCIALG_3:11;
end;
