reconsider p1 = p as primary Ideal of A ;
sqrt p1 = p by TOPZARI1:25, TOPZARI1:20;
then p1 is p -primary ;
then A1: p1 in { I where I is primary Ideal of A : I is p -primary } ;
{ I where I is primary Ideal of A : I is p -primary } c= PRIMARY A

proof

set X = { I where I is primary Ideal of A : I is p -primary } ;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { I where I is primary Ideal of A : I is p -primary } or x in PRIMARY A )
assume x in { I where I is primary Ideal of A : I is p -primary } ; :: thesis: x in PRIMARY A
then consider x1 being primary Ideal of A such that
A2: x1 = x and
x1 is p -primary ;
thus x in PRIMARY A by A2; :: thesis: verum

end;

hence { I where I is primary Ideal of A : I is p -primary } is non empty Subset of (PRIMARY A) by A1; :: thesis: verum