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 Th53:
for F being Field, E being F-homomorphic Field, K being Subfield of F
for f being Function of F,E, g being Function of K,E st
g = f|(the carrier of K) & f is additive
holds g is additive
proof
let F be Field,
    E be F-homomorphic Field,
    K be Subfield of F,
    f be Function of F,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 F by EC_PF_1:def 1;
  then reconsider x1 = x, y1 = y as Element of F;
A3: x + y = ((the addF of F)||the carrier of K).(x,y) by EC_PF_1:def 1
   .= 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;
