let X be BCI-Algebra_with_Condition(S); :: thesis: the ExternalDiff of X is associative

now

let a, b, c be Element of X; :: thesis: the ExternalDiff of X . (a,( the ExternalDiff of X . (b,c))) = the ExternalDiff of X . (( the ExternalDiff of X . (a,b)),c)
thus the ExternalDiff of X . (a,( the ExternalDiff of X . (b,c))) = a * (b * c)
.= (a * b) * c by Th10
.= the ExternalDiff of X . (( the ExternalDiff of X . (a,b)),c) ; :: thesis: verum

end;

hence the ExternalDiff of X is associative by BINOP_1:def 3; :: thesis: verum