set M = addLoopStr(# I,(add| (I,R)),(In ((0. R),I)) #);
reconsider M = addLoopStr(# I,(add| (I,R)),(In ((0. R),I)) #) as non empty addLoopStr ;
now :: thesis: for u being object st u in [:I,I:] holds
u in dom the addF of R
let u be object ; :: thesis: ( u in [:I,I:] implies u in dom the addF of R )
A1: dom the addF of R = [: the carrier of R, the carrier of R:] by FUNCT_2:def 1;
assume u in [:I,I:] ; :: thesis: u in dom the addF of R
hence u in dom the addF of R by A1; :: thesis: verum
end;
then [:I,I:] c= dom the addF of R ;
then A2: dom ( the addF of R || I) = [:I,I:] by RELAT_1:62;
A3: for a, b being Element of M
for a9, b9 being Element of I st a9 = a & b9 = b holds
a + b = a9 + b9
proof
let a, b be Element of M; :: thesis: for a9, b9 being Element of I st a9 = a & b9 = b holds
a + b = a9 + b9
let a9, b9 be Element of I; :: thesis: ( a9 = a & b9 = b implies a + b = a9 + b9 )
assume A4: ( a9 = a & b9 = b ) ; :: thesis: a + b = a9 + b9
[a9,b9] in dom ( the addF of R || I) by A2;
hence a + b = a9 + b9 by A4, FUNCT_1:47; :: thesis: verum
end;
now :: thesis: for a, b being Element of M holds a + b = b + a
let a, b be Element of M; :: thesis: a + b = b + a
reconsider a9 = a, b9 = b as Element of I ;
thus a + b = a9 + b9 by A3
.= b + a by A3 ; :: thesis: verum
end;
hence Gr (I,R) is Abelian by RLVECT_1:def 2; :: thesis: verum