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

theorem Th52:
  (for x,y,z being Element of X st (x\y)\z in I & y\z in I holds x
\z in I) implies for x,y,z being Element of X st (x\y)\z in I holds (x\z)\(y\z)
  in I
proof
  assume
A1: for x,y,z being Element of X st (x\y)\z in I & y\z in I holds x\z in I;
  let x,y,z be Element of X;
  x\(y\z) \ (x\y) \ (y\(y\z)) =0.X by BCIALG_1:1;
  then x\(y\z) \ (x\y)<=y\(y\z);
  then
A2: x\(y\z) \ (x\y)\z<=y\(y\z)\z by BCIALG_1:5;
  y\(y\z)\z =y\z\(y\z) by BCIALG_1:7;
  then x\(y\z) \ (x\y)\z<=0.X by A2,BCIALG_1:def 5;
  then x\(y\z) \ (x\y)\z\0.X=0.X;
  then x\(y\z) \ (x\y)\z=0.X by BCIALG_1:2;
  then
A3: (x\(y\z))\(x\y)\z in I by BCIALG_1:def 18;
  assume (x\y)\z in I;
  then x\(y\z)\z in I by A1,A3;
  hence thesis by BCIALG_1:7;
end;
