reserve i,j for Nat;
reserve A,B for Ring;

theorem Th12:
  for x1,x2 be Element of A st A is Subring of B holds
  In(x1,B) + In(x2,B) = In(x1+x2,B)
  proof
   let x1,x2 be Element of A;
    assume
A0: A is Subring of B; then
    x1 is Element of B by Lm6; then
A2: In(x1,B) = x1 by SUBSET_1:def 8;
    x2 is Element of B by A0,Lm6; then
A4: In(x2,B) = x2 by SUBSET_1:def 8;
    x1 + x2 is Element of B by A0,Lm6; then
    In(x1+x2,B) = x1+x2 by SUBSET_1:def 8
    .= In(x1,B)+In(x2,B) by A0,A2,A4,Th8;
    hence thesis;
   end;
