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 Th73:
  X is quasi-associative iff for x,y being Element of X holds x`\y =(x\y)`
proof
  thus X is quasi-associative implies for x,y being Element of X holds x`\y=(x
  \y)`
  proof
    assume X is quasi-associative;
    then for x,y being Element of X holds (x\y)`=(y\x)` by Th72;
    hence thesis by Lm16;
  end;
  assume for x,y being Element of X holds x`\y=(x\y)`;
  then for x,y being Element of X holds (x\y)\(y\x) in BCK-part(X)by Lm17;
  hence thesis by Lm18;
end;
