reserve A,B,C for non empty set,
  f for Function of [:A,B:],C;
reserve K for non empty doubleLoopStr;

theorem
  (for K being unital non empty doubleLoopStr holds 1.K = 1.opp(K)) &
  for K being add-associative right_zeroed right_complementable non empty
doubleLoopStr holds 0.K = 0.opp(K) & for x,y,z,u being (Scalar of K), a,b,c,d
being Scalar of opp(K) st x = a & y = b & z = c & u = d holds x+y = a+b & x*y =
b*a & -x = -a & (x+y)+z = (a+b)+c & x+(y+z) = a+(b+c) & (x*y)*z = c*(b*a) & x*(
  y*z) = (c*b)*a & x*(y+z) = (b+c)*a & (y+z)*x = a*(b+c) & x*y+z*u = b*a+d*c
by Lm3,Lm4;
