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 Th71:
  X is quasi-associative iff for x being Element of X holds x`<=x
proof
  thus X is quasi-associative implies for x being Element of X holds x`<=x
  proof
    assume
A1: X is quasi-associative;
    let x be Element of X;
    (x`)`\x=0.X by Th1;
    then x`\x=0.X by A1;
    hence thesis;
  end;
  assume for x being Element of X holds x`<=x;
  then for x,y being Element of X holds (x\y)`=(y\x)` by Lm15;
  then for x,y being Element of X holds x`\y=(x\y)` by Lm16;
  then for x,y being Element of X holds (x\y)\(y\x) in BCK-part(X)by Lm17;
  hence thesis by Lm18;
end;
