now :: thesis: for a, b, c being Element of doubleLoopStr(# the carrier of L, the addF of L, the multF of L, the OneF of L, the ZeroF of L #) holds (a + b) + c = a + (b + c)

let a, b, c be Element of doubleLoopStr(# the carrier of L, the addF of L, the multF of L, the OneF of L, the ZeroF of L #); :: thesis: (a + b) + c = a + (b + c)
reconsider x = a, y = b, z = c as Element of L ;
thus (a + b) + c = (x + y) + z
.= x + (y + z) by RLVECT_1:def 3
.= a + (b + c) ; :: thesis: verum

end;

hence doubleLoopStr(# the carrier of L, the addF of L, the multF of L, the OneF of L, the ZeroF of L #) is add-associative ; :: thesis: verum