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 )
assume A1: u in [:I,I:] ; :: thesis: u in dom the addF of R
dom the addF of R = [:the carrier of R,the carrier of R:] by FUNCT_2:def 1;
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 a', b' being Element of I st a' = a & b' = b holds
a + b = a' + b'

proof

let a, b be Element of M; :: thesis: for a', b' being Element of I st a' = a & b' = b holds
a + b = a' + b'
let a', b' be Element of I; :: thesis: ( a' = a & b' = b implies a + b = a' + b' )
A4: [a',b'] in dom (the addF of R || I) by A2;
assume ( a' = a & b' = b ) ; :: thesis: a + b = a' + b'
hence a + b = a' + b' by A4, FUNCT_1:70; :: thesis: verum

end;

reconsider I = I as RightIdeal of R ;
M is right_complementable

proof

let a be Element of M; :: according to ALGSTR_0:def 16 :: thesis: a is right_complementable
reconsider a' = a as Element of I ;
reconsider b = - a' as Element of M by Th14;
a + b = a' + (- a') by A3
.= 0. R by RLVECT_1:16
.= 0. M by Th3, FUNCT_7:def 1 ;
hence ex b being Element of M st a + b = 0. M ; :: according to ALGSTR_0:def 11 :: thesis: verum

end;

hence Gr I,R is right_complementable ; :: thesis: verum