 reserve R for commutative Ring;
 reserve A,B for non degenerated commutative Ring;
 reserve h for Function of A,B;
 reserve I0,I,I1,I2 for Ideal of A;
 reserve J,J1,J2 for proper Ideal of A;
 reserve p for prime Ideal of A;
 reserve S,S1 for non empty Subset of A;
 reserve E,E1,E2 for Subset of A;
 reserve a,b,f for Element of A;
 reserve n for Nat;
 reserve x,o,o1 for object;

theorem Lm5:
  not 1.A in sqrt J
  proof
    assume 1.A in sqrt J; then
    1.A in {a where a is Element of A: ex n be Element of NAT st a|^n in J}
      by IDEAL_1:def 24; then
    consider a be Element of A such that
A3: 1.A = a and
A4: ex n be Element of NAT st a|^n in J;
    consider n1 be Element of NAT such that
A5: a|^n1 in J by A4;
    1.A = a|^n1 by A3,Lm4;
    hence contradiction by IDEAL_1:19, A5;
  end;
