reserve R for Ring,
  I for Ideal of R,
  a, b for Element of R;
reserve R for add-associative right_zeroed right_complementable Abelian
    distributive left_unital non empty doubleLoopStr;
reserve I for Ideal of R;
reserve a,b for Element of R;
reserve x, y for Element of R/I;

theorem Th13:
  x = Class(EqRel(R,I),a) & y = Class(EqRel(R,I),b) implies
  x+y = Class(EqRel(R,I),a+b)
proof
  consider a1, b1 being Element of R such that
A1: x = Class(EqRel(R,I),a1) & y = Class(EqRel(R,I),b1) and
A2: (the addF of R/I).(x,y) = Class(EqRel(R,I),a1+b1) by Def6;
A3: a1-a+(b1-b) = a1-a+b1-b by RLVECT_1:28
    .= a1+b1-a-b by RLVECT_1:28
    .= a1+b1-(a+b) by RLVECT_1:27;
  assume x = Class(EqRel(R,I),a) & y = Class(EqRel(R,I),b);
  then a1-a in I & b1-b in I by A1,Th6;
  then a1+b1-(a+b) in I by A3,IDEAL_1:def 1;
  hence thesis by A2,Th6;
end;
