let V be non empty add-associative addLoopStr ; for M1, M2, M3 being Subset of V holds (M1 + M2) + M3 = M1 + (M2 + M3)
let M1, M2, M3 be Subset of V; (M1 + M2) + M3 = M1 + (M2 + M3)
for x being Element of V st x in M1 + (M2 + M3) holds
x in (M1 + M2) + M3
proof
let x be
Element of
V;
( x in M1 + (M2 + M3) implies x in (M1 + M2) + M3 )
assume
x in M1 + (M2 + M3)
;
x in (M1 + M2) + M3
then
x in { (u + v) where u, v is Element of V : ( u in M1 & v in M2 + M3 ) }
by RUSUB_4:def 9;
then consider x1,
x9 being
Element of
V such that A1:
x = x1 + x9
and A2:
x1 in M1
and A3:
x9 in M2 + M3
;
x9 in { (u + v) where u, v is Element of V : ( u in M2 & v in M3 ) }
by A3, RUSUB_4:def 9;
then consider x2,
x3 being
Element of
V such that A4:
x9 = x2 + x3
and A5:
x2 in M2
and A6:
x3 in M3
;
x1 + x2 in { (u + v) where u, v is Element of V : ( u in M1 & v in M2 ) }
by A2, A5;
then A7:
x1 + x2 in M1 + M2
by RUSUB_4:def 9;
x = (x1 + x2) + x3
by A1, A4, RLVECT_1:def 3;
then
x in { (u + v) where u, v is Element of V : ( u in M1 + M2 & v in M3 ) }
by A6, A7;
hence
x in (M1 + M2) + M3
by RUSUB_4:def 9;
verum
end;
then A8:
M1 + (M2 + M3) c= (M1 + M2) + M3
;
for x being Element of V st x in (M1 + M2) + M3 holds
x in M1 + (M2 + M3)
proof
let x be
Element of
V;
( x in (M1 + M2) + M3 implies x in M1 + (M2 + M3) )
assume
x in (M1 + M2) + M3
;
x in M1 + (M2 + M3)
then
x in { (u + v) where u, v is Element of V : ( u in M1 + M2 & v in M3 ) }
by RUSUB_4:def 9;
then consider x9,
x3 being
Element of
V such that A9:
x = x9 + x3
and A10:
x9 in M1 + M2
and A11:
x3 in M3
;
x9 in { (u + v) where u, v is Element of V : ( u in M1 & v in M2 ) }
by A10, RUSUB_4:def 9;
then consider x1,
x2 being
Element of
V such that A12:
x9 = x1 + x2
and A13:
x1 in M1
and A14:
x2 in M2
;
x2 + x3 in { (u + v) where u, v is Element of V : ( u in M2 & v in M3 ) }
by A11, A14;
then A15:
x2 + x3 in M2 + M3
by RUSUB_4:def 9;
x = x1 + (x2 + x3)
by A9, A12, RLVECT_1:def 3;
then
x in { (u + v) where u, v is Element of V : ( u in M1 & v in M2 + M3 ) }
by A13, A15;
hence
x in M1 + (M2 + M3)
by RUSUB_4:def 9;
verum
end;
then
(M1 + M2) + M3 c= M1 + (M2 + M3)
;
hence
(M1 + M2) + M3 = M1 + (M2 + M3)
by A8; verum