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 Th8:
    I ||^ 1 = I & I *' [#]A = I
    proof
      set f = <*I*>;
A1:   len f = 1 & f.1 = I by FINSEQ_1:40;
A2:  for i be Nat st i in dom f & i+1 in dom f holds f.(i+1) = I *' (f/.i)
      proof
        let i be Nat;
        assume
A3:     i in dom f & i+1 in dom f;
        dom f = {1} by FINSEQ_1:2,38; then
        i = 1 & i+1 = 1 by A3,TARSKI:def 1;
        hence thesis;
      end;
      I + [#]A = [#]A by IDEAL_1:74; then
      I, [#]A are_co-prime; then
      I *' [#]A  = I /\ [#]A by IDEAL_1:84 .= I by XBOOLE_1:28;
      hence thesis by A1,A2,Def2;
    end;
