let L be non empty addLoopStr ; :: thesis: ( L is AddGroup iff ( ( for a being Element of L holds a + (0. L) = a ) & ( for a being Element of L ex x being Element of L st a + x = 0. L ) & ( for a, b, c being Element of L holds (a + b) + c = a + (b + c) ) ) )
thus ( L is AddGroup implies ( ( for a being Element of L holds a + (0. L) = a ) & ( for a being Element of L ex x being Element of L st a + x = 0. L ) & ( for a, b, c being Element of L holds (a + b) + c = a + (b + c) ) ) ) by Th7, RLVECT_1:def 6, RLVECT_1:def 7; :: thesis: ( ( for a being Element of L holds a + (0. L) = a ) & ( for a being Element of L ex x being Element of L st a + x = 0. L ) & ( for a, b, c being Element of L holds (a + b) + c = a + (b + c) ) implies L is AddGroup )
assume that
Z1: for a being Element of L holds a + (0. L) = a and
Z2: for a being Element of L ex x being Element of L st a + x = 0. L and
Z3: for a, b, c being Element of L holds (a + b) + c = a + (b + c) ; :: thesis: L is AddGroup
L is right_complementable

proof

let a be Element of L; :: according to ALGSTR_0:def 16 :: thesis: a is right_complementable
thus ex x being Element of L st a + x = 0. L by Z2; :: according to ALGSTR_0:def 11 :: thesis: verum

end;

hence L is AddGroup by Z1, Z3, RLVECT_1:def 6, RLVECT_1:def 7; :: thesis: verum