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 Th37:
    for q be proper Ideal of A holds
    (canHom q).:(sqrt q) = nilrad(A/q)
    proof
      let q be proper Ideal of A;
A1:   for x be object st x in nilrad (A/q) holds x in (canHom q).:(sqrt q)
      proof
        let x be object;
        assume x in nilrad (A/q); then
        x in sqrt({0.(A/q)}) by TOPZARI1:17; then
        x in {a where a is Element of A/q: ex n being
        Element of NAT st a|^n in {0.(A/q)}} by IDEAL_1:def 24; then
        consider x1 be Element of A/q such that
A3:     x1 = x and
A4:     ex n being Element of NAT st x1|^n in {0.(A/q)};
        consider n1 be Element of NAT such that
A5:     x1|^n1 in {0.(A/q)} by A4;
        consider y1 being Element of A such that
A6:     x1 = Class(EqRel(A,q),y1) by RING_1:11;
A7:     x1 = (canHom q).y1 by A6,RING_2:def 5;
        Class(EqRel(A,q),0.A) = 0.(A/q) by RING_1:def 6
        .= x1|^n1 by A5,TARSKI:def 1
        .= Class(EqRel(A,q),y1|^n1) by A6,FIELD_1:2; then
        y1|^n1 - 0.A in q by RING_1:6; then
        y1 in {a where a is Element of A:
               ex n being Element of NAT st a|^n in q}; then
A8:     y1 in sqrt q by IDEAL_1:def 24;
        dom canHom q = [#]A by FUNCT_2:def 1;
        hence thesis by A3,FUNCT_1:def 6,A7,A8;
      end;
      (canHom q).:(sqrt q) c= nilrad (A/q)
      proof
        let x be object;
        assume x in (canHom q).:(sqrt q); then
        consider x1 be object such that
A11:    x1 in dom canHom q and
A12:    x1 in sqrt q and
A13:    x = (canHom q).x1 by FUNCT_1:def 6;
        reconsider x1 as Element of A by A11;
A14:    x1 in {a where a is Element of A:
               ex n being Element of NAT st a|^n in q} by A12,IDEAL_1:def 24;
        consider x0 be Element of A such that
A15:    x0 = x1 and
A16:    ex n being Element of NAT st x0|^n in q by A14;
        consider n1 be Element of NAT such that
A17:    x0|^n1 in q by A16;
A18:    x0|^n1 - 0.A in q by A17;
A19:    x = Class(EqRel(A,q),x0) by A13,A15,RING_2:def 5;
        x in rng (canHom q) by A11, A13,FUNCT_1:def 3; then
        reconsider y = x as Element of A/q;
        y|^n1 = Class(EqRel(A,q),x0|^n1) by A19,FIELD_1:2
        .= Class(EqRel(A,q),0.A) by A18,RING_1:6 .= 0.(A/q) by RING_1:def 6;
        then
        y|^n1 in {0.(A/q)} by TARSKI:def 1; then
        y in {a where a is Element of A/q: ex n being
        Element of NAT st a|^n in {0.(A/q)}}; then
        y in sqrt {0.(A/q)} by IDEAL_1:def 24;
        hence thesis by TOPZARI1:17;
      end;
      hence thesis by A1,TARSKI:2;
    end;
