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 Th17:
    for x,y be Element of the carrier of A/q st
    x in (canHom(q)).:I & y in (canHom(q)).:I holds x + y in (canHom(q)).:I
    proof
      let x,y be Element of the carrier of A/q such that
A1:   x in (canHom(q)).:I and
A2:   y in (canHom(q)).:I;
      consider x0 being object such that
A3:   x0 in dom(canHom(q)) and
A4:   x0 in I and
A5:   x = (canHom(q)).x0 by A1,FUNCT_1:def 6;
      consider y0 being object such that
A6:   y0 in dom(canHom(q)) and
A7:   y0 in I and
A8:   y = (canHom(q)).y0 by A2,FUNCT_1:def 6;
A9:   dom(canHom(q)) = the carrier of A by FUNCT_2:def 1;
      reconsider x1= x0,y1=y0 as Element of A by A3,A6;
A10:  (canHom q).x1 = Class(EqRel(A,q),x1) by RING_2:def 5;
A11:  (canHom q).y1 = Class(EqRel(A,q),y1) by RING_2:def 5;
A12:  x1+y1 in I by A4,A7,IDEAL_1:def 1;
      (canHom(q)).(x1+y1) = Class(EqRel(A,q),x1+y1) by RING_2:def 5
      .= (canHom q).x1 + (canHom q).y1 by A10,A11,RING_1:13;
      hence thesis by A5,A8,A9,FUNCT_1:def 6,A12;
    end;
