 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;
 reserve m for maximal Ideal of A;
 reserve p for prime Ideal of A;
 reserve k for non zero Nat;

theorem Th40:
  I1 *' I2 c= p implies I1 c= p or I2 c= p
   proof
     not(I1 c= p or I2 c= p) implies not(I1 *' I2 c= p)
     proof
       assume
A1:    not (I1 c= p or I2 c= p); then
       consider x1 be object such that
A2:    x1 in I1 and
A3:    not x1 in p;
       consider x2 be object such that
A4:    x2 in I2 and
A5:    not x2 in p by A1;
       reconsider x1 as Element of A by A2;
       reconsider x2 as Element of A by A4;
       reconsider x = x1*x2 as Element of the carrier of A;
       reconsider seq=<* x *> as FinSequence of the carrier of A;
A6:    Sum seq = x by BINOM:3;
A8:    for i being Element of NAT st 1 <= i & i <= len seq
       ex a,b being Element of A st seq.i = a*b & a in I1 & b in I2
       proof
         let i be Element of NAT;
         assume that
A9:      1 <= i and
A10:     i <= len seq;
A11:     i <= 1 by A10,FINSEQ_1:40;
         seq.i = seq.1 by A9,XXREAL_0:1, A11 .= x1*x2 by FINSEQ_1:40;
         hence thesis by A2,A4;
       end;
A12:   Sum seq in
       {Sum s where s is FinSequence of the carrier of A : for i being
        Element of NAT st 1 <= i & i <= len s ex a,b be Element of A
        st s.i = a*b & a in I1 & b in I2} by A8;
       x1*x2 in I1 *' I2 by A6,A12,IDEAL_1:def 21;
       hence thesis by A3,A5,RING_1:def 1;
     end;
     hence thesis;
   end;
