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 Th72:
  X is quasi-associative iff for x,y being Element of X holds (x\y )`=(y\x)`
proof
  thus X is quasi-associative implies for x,y being Element of X holds (x\y)`=
  (y\x)`
  proof
    assume X is quasi-associative;
    then for x being Element of X holds x`<=x by Th71;
    hence thesis by Lm15;
  end;
  assume for x,y being Element of X holds (x\y)`=(y\x)`;
  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;
