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

let u be set ; :: 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 by TARSKI:def 3;
then A2: dom (the addF of R || I) = [:I,I:] by RELAT_1:91;
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_1:70; :: thesis: verum

end;

now

let a, b, c be Element of M; :: thesis: (a + b) + c = a + (b + c)
reconsider a9 = a, b9 = b, c9 = c as Element of I ;
a9 + b9 in I by Def1;
then A5: [(a9 + b9),c9] in dom (the addF of R || I) by A2, ZFMISC_1:def 2;
b9 + c9 in I by Def1;
then A6: [a9,(b9 + c9)] in dom (the addF of R || I) by A2, ZFMISC_1:def 2;
thus (a + b) + c = (the addF of R || I) . [(a9 + b9),c9] by A3
.= (a9 + b9) + c9 by A5, FUNCT_1:70
.= a9 + (b9 + c9) by RLVECT_1:def 6
.= (add| I,R) . [a9,(b9 + c9)] by A6, FUNCT_1:70
.= a + (b + c) by A3 ; :: thesis: verum

end;

hence Gr I,R is add-associative by RLVECT_1:def 6; :: thesis: verum