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 Th68:
for R being Ring,
    S being R-homomorphic Ring, T being S-homomorphic Ring
for f being Homomorphism of R,S
for g being Homomorphism of S,T holds ker(f) c= ker(g*f)
proof
let R be Ring, S be R-homomorphic Ring, T be S-homomorphic Ring;
let f be Homomorphism of R,S; let g be Homomorphism of S,T;
now let x be object;
  assume x in ker f;
  then consider r being Element of R such that A1: x = r & f.r = 0.S;
  (g*f).r = g.(f.r) by FUNCT_2:15 .= 0.T by A1,RING_2:6;
  hence x in ker(g*f) by A1;
  end;
hence thesis;
end;
