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 Th48:
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 additive
holds g is additive
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 additive;
  let x,y be Element of K;
  the carrier of K c= the carrier of R by C0SP1:def 3;
  then reconsider x1=x, y1=y as Element of R;
A3: x + y = ((the addF of R)||the carrier of K).(x,y) by C0SP1:def 3
   .= x1 + y1 by Th1;
   thus g.(x+y) = f.(x+y) by A1,FUNCT_1:49
   .= f.x1 + f.y1 by A2,A3
   .= g.x + f.y1 by A1,FUNCT_1:49
   .= g.x + g.y by A1,FUNCT_1:49;
end;
