reserve X for BCI-algebra;
reserve x,y,z,u,a,b for Element of X;
reserve IT for non empty Subset of X;

theorem Th50:
  for X being non empty BCIStr_0 holds (X is associative
  BCI-algebra iff for x,y,z being Element of X holds (x\y)\(x\z)=z\y & x`=x )
proof
  let X be non empty BCIStr_0;
  thus X is associative BCI-algebra implies for x,y,z being Element of X holds
  (x\y)\(x\z)=z\y & x`=x
  proof
    assume
A1: X is associative BCI-algebra;
    let x,y,z be Element of X;
    (y\x)\(z\x)=z\y by A1,Th49;
    then
A2: (x\y)\(z\x)=z\y by A1,Th48;
    x\0.X=x by A1,Th49;
    hence thesis by A1,A2,Th48;
  end;
  assume
A3: for x,y,z being Element of X holds (x\y)\(x\z)=z\y & x`=x;
  for x,y,z being Element of X holds (y\x)\(z\x)=z\y & x\0.X=x
  proof
A4: for x,y being Element of X holds y\x=x\y
    proof
      let x,y be Element of X;
      y`\x`=x\y by A3;
      then y\x`=x\y by A3;
      hence thesis by A3;
    end;
    let x,y,z be Element of X;
A5: x`=x by A3;
    (x\y)\(x\z)=z\y by A3;
    then (y\x)\(x\z)=z\y by A4;
    hence thesis by A4,A5;
  end;
  hence thesis by Th49;
end;
