theorem :: MOD_4:3

( ( for K being non empty unital doubleLoopStr holds 1. K = 1. (opp K) ) & ( for K being non empty right_complementable add-associative right_zeroed doubleLoopStr holds
( 0. K = 0. (opp K) & ( for x, y, z, u being Scalar of K
for 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;