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 Th19:
    (canHom q).:I is Ideal of A/q
    proof
A1:   (canHom q).:I is add-closed by Th17;
A2:   (canHom q).:I is left-ideal by Th18;
A3:   0.A in I by IDEAL_1:2;
A4:   (canHom q).0.A =  Class(EqRel(A,q),0.A) by RING_2:def 5
      .= 0.(A/q) by RING_1:def 6;
      0.A in the carrier of A; then
      0.A in dom(canHom q) by FUNCT_2:def 1; then
      0.(A/q) in (canHom q).:I by A3,A4,FUNCT_1:def 6;
      hence thesis by A1,A2;
    end;
