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
  for x,y being Element of X holds (0.X\(x\y))\(y\x)=0.X
proof
  let x,y be Element of X;
  ((0.X\x)\(0.X\y))\(y\x)=((0.X\x)\(y\x))\(0.X\y) by BCIALG_1:7;
  then
A1: ((0.X\x)\(0.X\y))\(y\x)=(((y\x)`)\x)\(0.X\y) by BCIALG_1:7
    .=(((y`\x`))\x)\(0.X\y) by BCIALG_1:9
    .=(((y`\x))\x`)\(0.X\y) by BCIALG_1:7
    .=(((y`\x))\y`)\x` by BCIALG_1:7
    .=(((y`\y`))\x)\x` by BCIALG_1:7
    .=x`\x` by BCIALG_1:def 5
    .=0.X by BCIALG_1:def 5;
  (0.X\(x\y))\(y\x)=((x\y)`)\(y\x) .=(x`\y`)\(y\x) by BCIALG_1:9
    .=((0.X\x)\(0.X\y))\(y\x);
  hence thesis by A1;
end;
