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
     sqrt(I),sqrt(J) are_co-prime implies I,J are_co-prime
     proof
       assume
A1:    sqrt(I),sqrt(J) are_co-prime;
       sqrt (I +J) = sqrt( sqrt(I)+ sqrt(J)) by Th14 .= [#]A by A1,Th13;
       hence thesis by Th13;
     end;
