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 Th21:
    for r,x be Element of the carrier of A st x in (canHom q)"M1
    holds r*x in (canHom q)"M1
    proof
      let r,x be Element of the carrier of A such that
A1:   x in (canHom q)"M1;
A2:   (canHom q).x in M1 by A1,FUNCT_1:def 7;
      reconsider a=x,b=r as Element of A;
reconsider x1= Class(EqRel(A,q),a) as Element of A/q by RING_1:12;
reconsider r1= Class(EqRel(A,q),b) as Element of A/q by RING_1:12;
      (canHom q).r * (canHom q).x =
      r1 * (canHom q).x by RING_2:def 5 .= r1 * x1 by RING_2:def 5
      .= Class(EqRel(A,q),a*b) by RING_1:14; then
      Class(EqRel(A,q),a*b) in M1 by A2,IDEAL_1:def 2; then
A4:   (canHom q).(a*b) in M1 by RING_2:def 5;
      a*b in A; then
      a*b in dom (canHom q) by FUNCT_2:def 1;
      hence thesis by A4,FUNCT_1:def 7;
    end;
