reserve p,q for Rational;
reserve g,m,m1,m2,n,n1,n2 for Nat;
reserve i,i1,i2,j,j1,j2 for Integer;
reserve R for Ring, F for Field;

theorem Th69:
for R being Ring,
    S being R-homomorphic Ring, T being S-monomorphic Ring
for f being Homomorphism of R,S
for g being Monomorphism of S,T holds ker(f) = ker(g*f)
proof
let R be Ring, S be R-homomorphic Ring, T be S-monomorphic Ring;
let f be Homomorphism of R,S; let g be Monomorphism of S,T;
A1: ker g = {0.S} by RING_2:12;
now let x be object;
  assume x in ker(g*f);
  then consider r being Element of R such that A2: x = r & (g*f).r = 0.T;
  g.(f.r) = 0.T by A2,FUNCT_2:15;
  then f.r in {0.S} by A1;
  then f.r = 0.S by TARSKI:def 1;
  hence x in ker f by A2;
  end;
then ker(g*f) c= ker f;
hence thesis by Th68;
end;
