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 Th25:
    q c= I implies (canHom q)"((canHom q).:I) = I
    proof
      assume q c= I; then
      ker(canHom q) = q & q c= I by RING_2:13; then
A2:   I + ker(canHom q) c= I by IDEAL_1:75;
      (canHom q)"((canHom q).:I) c= I
      proof
        for x be object st x in (canHom q)"((canHom q).:I) holds x in I
        proof
          set f = canHom q;
          let x be object;
          assume
A4:       x in f"(f.:I); then
A5:       x in dom f & f.x in (f.:I) by FUNCT_1:def 7;
          consider x1 be object such that
A6:       x1 in dom f and
A7:       x1 in I and
A8:       f.x = f.x1 by A5,FUNCT_1:def 6;
          reconsider a=x,b=x1 as Element of A by A4,A6;
          f.a - f.b = 0.(A/q) by A8,RLVECT_1:5; then
          f.(a-b) = 0.(A/q) by RING_2:8; then
          (a-b) in {v where v is Element of A : f.v = 0.(A/q)}; then
          (a-b) in ker f by VECTSP10:def 9; then
          consider x3 be object such that
A10:      x3 in ker f and
A11:      x3 = a-b;
          reconsider c = x3 as Element of A by A10;
A12:      a = b + c by A11, VECTSP_2:2;
          I+ker f={a + b where a,b is Element of A : a in I & b in ker f}
            by IDEAL_1:def 19; then
          x in I + ker f by A12,A7,A10;
          hence thesis by A2;
        end;
        hence thesis;
      end;
      hence thesis by FUNCT_2:42;
    end;
