
theorem e1:
for R1,R2,S2 being Ring,
    S1 being RingExtension of R1
for h1 being Function of R1,R2, h2 being Function of S1,S2
holds h2 is h1-extending iff h2|R1 = h1
proof
let R1,R2,S2 be Ring, S1 be RingExtension of R1;
let h1 be Function of R1,R2, h2 be Function of S1,S2;
H: R1 is Subring of S1 by FIELD_4:def 1;
now assume AS: h2 is h1-extending;
  A0: dom h2 = the carrier of S1 &
      dom h1 = the carrier of R1 by FUNCT_2:def 1; 
  the carrier of R1 c= the carrier of S1 by H,C0SP1:def 3; then
  A1: dom h1 = dom h2 /\ (the carrier of R1) by A0,XBOOLE_1:28;
  for x being object st x in dom h1 holds h2.x = h1.x by AS;
  hence h2|R1 = h1 by A1,FUNCT_1:46;
  end;
hence thesis by FUNCT_1:49;
end;
