reserve R for commutative Ring;
reserve A for non degenerated commutative Ring;
reserve I,J,q for Ideal of A;
reserve p for prime Ideal of A;
reserve M,M1,M2 for Ideal of A/q;

theorem Th20:
    for x,y be Element of the carrier of A st x in (canHom q)"M1  &
    y in (canHom q)"M1 holds x + y in (canHom q)"M1
    proof
      let x,y be Element of the carrier of A such that
A1:   x in (canHom q)"M1 and
A2:   y in (canHom q)"M1;
A3:   (canHom q).x in M1 by A1,FUNCT_1:def 7;
A4:   (canHom q).y in M1 by A2,FUNCT_1:def 7;
      reconsider a=x,b=y as Element of A;
reconsider x1= Class(EqRel(A,q),a) as Element of A/q by RING_1:12;
reconsider y1= Class(EqRel(A,q),b) as Element of A/q by RING_1:12;
      (canHom q).x + (canHom q).y =
      x1 + (canHom q).y by RING_2:def 5 .= x1 + y1 by RING_2:def 5
      .= Class(EqRel(A,q),a+b) by RING_1:13; then
A5:   Class(EqRel(A,q),a+b) in M1 by A3,A4,IDEAL_1:def 1;
A6:   (canHom q).(a+b) in M1 by A5,RING_2:def 5;
      a+b in A; then
      a+b in dom (canHom q) by FUNCT_2:def 1;
      hence thesis by A6, FUNCT_1:def 7;
    end;
