now :: thesis: for a, b 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 = b + a

let a, b 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 = b + a
reconsider x = a, y = b as Element of L ;
thus a + b = x + y
.= y + x
.= b + a ; :: 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 Abelian ; :: thesis: verum