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 Th30:
    q c= I & I is maximal implies (canHom q).:I is maximal
    proof
      set M = (canHom q).:I;
      reconsider M as Ideal of A/q by Th19;
      assume
A1:   q c= I & I is maximal;
      (canHom q).:I is maximal
      proof
        assume not (canHom q).:I is maximal; then
        per cases;
          suppose
            not M is proper; then
            M = [#](A/q); then
            [#]A = (canHom q)"M by Th28 .= I by A1,Th25;
            hence contradiction by A1;
          end;
          suppose
            not (M is quasi-maximal); then
            consider J be Ideal of A/q such that
A6:         M c= J & J <> M & J is proper;
A7:         (canHom q)"M <> (canHom q)"J by Lm2,A6;
A8:         (canHom q)"J is proper by A6,Th29;
            reconsider I2 = (canHom q)"J as Ideal of A by Th22;
            (canHom q)"M = I by Th25,A1; then
            consider I0 be Ideal of A such that
            I0 = I2 and
A9:         I c= I0 & I <> I0 & I0 is proper by A6, Th27, A7, A8;
            not I is quasi-maximal by A9;
            hence contradiction by A1;
          end;
        end;
        hence thesis;
      end;
