 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;
 reserve M0 for Ideal of B;
 reserve M0 for prime Ideal of B;

theorem Th51:
  h is RingHomomorphism implies h"M0 is prime Ideal of A
   proof
     assume
A1:  h is RingHomomorphism;
A2:  for x,y be Element of A st x*y in h"M0 holds x in h"M0 or y in h"M0
     proof
       let x,y be Element of A;
       assume x*y in h"M0; then
       x*y in dom(h) & h.(x*y) in M0 by FUNCT_1:def 7; then
A4:    h.x * h.y in M0 by A1,GROUP_6:def 6;
A5:    dom(h)= the carrier of A by FUNCT_2:def 1;
       x in h"M0 or y in h"M0
       proof
         per cases by A4,RING_1:def 1;
           suppose h.x in M0;
             hence thesis by A5,FUNCT_1:def 7;
           end;
           suppose h.y in M0;
             hence thesis by A5,FUNCT_1:def 7;
           end;
         end;
       hence thesis;
     end;
     h"M0 <> the carrier of A
     proof
       assume
A9:    h"M0 = the carrier of A;
A10:   h is unity-preserving by A1;
       1.B in M0 by A9,FUNCT_1:def 7,A10;
       hence contradiction by IDEAL_1:19;
     end; then
     h"M0 is proper quasi-prime by A2;
     hence thesis by A1,Th50;
   end;
