
theorem e0:
for R,S2 being Ring,
    S1 being RingExtension of R
for h being Function of S1,S2 holds h is R-fixing iff h|R = id R
proof
let R,S2 be Ring, S1 be RingExtension of R;
let h be Function of S1,S2;
H: R is Subring of S1 by FIELD_4:def 1;
A: now assume AS: h is R-fixing;
   A0: dom h = the carrier of S1 by FUNCT_2:def 1; 
   the carrier of R c= the carrier of S1 by H,C0SP1:def 3; then
   A1: dom(id R) = dom h /\ (the carrier of R) by A0,XBOOLE_1:28;
   now let x be object;
     assume x in dom(id R);
     then reconsider a = x as Element of R;
     h.a = a by AS;
     hence (id R).x = h.x;
     end;
   hence h|R = id R by A1,FUNCT_1:46;
   end;
now assume AS: h|R = id R;
  now let a be Element of R;
    thus h.a = (h|(the carrier of R)).a by FUNCT_1:49 
            .= a by AS;
    end;
  hence h is R-fixing;
  end;
hence thesis by A;
end;
