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 Th50:
for R being Ring, E being R-homomorphic Ring, K being Subring of R
for f being Function of R,E, g being Function of K,E st
g = f|(the carrier of K) & f is unity-preserving
holds g is unity-preserving
proof
let R be Ring,
    E be R-homomorphic Ring,
    K be Subring of R,
    f be Function of R,E,
    g be Function of K,E such that
A1: g = f|(the carrier of K) and
A2: f is unity-preserving;
  thus g.(1_K) = f.(1_K) by A1,FUNCT_1:49
  .= 1_E by A2,C0SP1:def 3;
end;
