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 Th18:
    for a,x be Element of the carrier of A/q st
    x in (canHom(q)).:I holds a*x in (canHom(q)).:I
    proof
      let a,x be Element of the carrier of A/q such that
A1:   x in (canHom(q)).:I;
A2:   rng(canHom(q)) = the carrier of A/q by FUNCT_2:def 3;
      consider a0 being object such that
A3:   a0 in dom(canHom(q)) and
A4:   a = (canHom(q)).a0 by A2,FUNCT_1:def 3;
      consider x0 being object such that
A5:   x0 in dom(canHom(q)) and
A6:   x0 in I and
A7:   x = (canHom(q)).x0 by A1,FUNCT_1:def 6;
A8:   dom(canHom(q)) = the carrier of A by FUNCT_2:def 1;
      reconsider a1 = a0 ,x1= x0 as Element of A by A5,A3;
A9:   (canHom q).a1 = Class(EqRel(A,q),a1) by RING_2:def 5;
A10:  (canHom q).x1 = Class(EqRel(A,q),x1) by RING_2:def 5;
A11:  a1*x1 in I by A6,IDEAL_1:def 2;
      (canHom(q)).(a1*x1) = Class(EqRel(A,q),a1*x1) by RING_2:def 5
      .= (canHom q).a1 * (canHom q).x1 by A9,A10,RING_1:14;
      hence thesis by A4, A7, A8,A11, FUNCT_1:def 6;
    end;
