for u1, u2, u3 being Element of (Prod_of_RLS (X,Y)) holds (u1 + u2) + u3 = u1 + (u2 + u3)
proof
let u1,
u2,
u3 be
Element of
(Prod_of_RLS (X,Y));
(u1 + u2) + u3 = u1 + (u2 + u3)
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;
consider x3,
y3 being
object such that A4:
x3 in the
carrier of
X
and A5:
y3 in the
carrier of
Y
and A6:
u3 = [x3,y3]
by ZFMISC_1:def 2;
reconsider y3 =
y3 as
VECTOR of
Y by A5;
reconsider x3 =
x3 as
VECTOR of
X by A4;
reconsider x1 =
x1 as
VECTOR of
X by A1;
consider x2,
y2 being
object such that A7:
x2 in the
carrier of
X
and A8:
y2 in the
carrier of
Y
and A9:
u2 = [x2,y2]
by ZFMISC_1:def 2;
reconsider y2 =
y2 as
VECTOR of
Y by A8;
reconsider x2 =
x2 as
VECTOR of
X by A7;
u1 + u2 = [(x1 + x2),(y1 + y2)]
by A3, A9, Def1;
then A10:
(u1 + u2) + u3 = [((x1 + x2) + x3),((y1 + y2) + y3)]
by A6, Def1;
u2 + u3 = [(x2 + x3),(y2 + y3)]
by A9, A6, Def1;
then A11:
u1 + (u2 + u3) = [(x1 + (x2 + x3)),(y1 + (y2 + y3))]
by A3, Def1;
(x1 + x2) + x3 = x1 + (x2 + x3)
by RLVECT_1:def 3;
hence
(u1 + u2) + u3 = u1 + (u2 + u3)
by A10, A11, RLVECT_1:def 3;
verum
end;
hence
Prod_of_RLS (X,Y) is add-associative
; verum