theorem Th48:
  (for x,y being Element of X holds y\x=x\y) iff X is associative
proof
  thus (for x,y being Element of X holds y\x=x\y) implies X is associative by
Lm6;
  assume X is associative;
  then for x being Element of X holds x`=x by Th47;
  hence thesis by Lm7;
end;
