for u1, u2 being Element of (Prod_of_RLS (X,Y)) holds u1 + u2 = u2 + u1
proof
let u1,
u2 be
Element of
(Prod_of_RLS (X,Y));
u1 + u2 = u2 + u1
consider x1,
y1 being
object such that A1:
x1 in the
carrier of
X
and A2:
y1 in the
carrier of
Y
and A3:
u1 = [x1,y1]
by ZFMISC_1:def 2;
reconsider y1 =
y1 as
VECTOR of
Y by A2;
reconsider x1 =
x1 as
VECTOR of
X by A1;
consider x2,
y2 being
object such that A4:
x2 in the
carrier of
X
and A5:
y2 in the
carrier of
Y
and A6:
u2 = [x2,y2]
by ZFMISC_1:def 2;
reconsider y2 =
y2 as
VECTOR of
Y by A5;
reconsider x2 =
x2 as
VECTOR of
X by A4;
u1 + u2 = [(x2 + x1),(y2 + y1)]
by A3, A6, Th2;
hence
u1 + u2 = u2 + u1
by A3, A6, Def1;
verum
end;
hence
Prod_of_RLS (X,Y) is Abelian
; verum