let F be Field; :: thesis: addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #) is AbGroup
set L = addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #);
A1: addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #) is add-associative
proof
let u, v, w be Element of addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #); :: according to RLVECT_1:def 3 :: thesis: (u + v) + w = u + (v + w)
reconsider a = u, b = v, c = w as Element of F ;
thus (u + v) + w = (a + b) + c
.= a + (b + c) by RLVECT_1:def 3
.= u + (v + w) ; :: thesis: verum
end;
A2: addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #) is right_zeroed
proof
let v be Element of addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #); :: according to RLVECT_1:def 4 :: thesis: v + (0. addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #)) = v
reconsider a = v as Element of F ;
thus v + (0. addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #)) = a + (0. F)
.= v ; :: thesis: verum
end;
A3: addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #) is right_complementable
proof
let v be Element of addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #); :: according to ALGSTR_0:def 16 :: thesis: v is right_complementable
reconsider a = v as Element of F ;
consider b being Element of F such that
A4: a + b = 0. F by ALGSTR_0:def 11;
reconsider w = b as Element of addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #) ;
take w ; :: according to ALGSTR_0:def 11 :: thesis: v + w = 0. addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #)
thus v + w = 0. addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #) by A4; :: thesis: verum
end;
addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #) is Abelian
proof
let v, w be Element of addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #); :: according to RLVECT_1:def 2 :: thesis: v + w = w + v
reconsider a = v, b = w as Element of F ;
thus v + w = a + b
.= b + a
.= w + v ; :: thesis: verum
end;
hence addLoopStr(# the carrier of F, the addF of F, the ZeroF of F #) is AbGroup by A1, A2, A3; :: thesis: verum