reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem
  x in dom f & x in X implies f.x in rng(f|X)
proof
  assume that
A1: x in dom f and
A2: x in X;
  x in dom f /\ X by A1,A2,XBOOLE_0:def 4;
  then
A3: x in dom(f|X) by RELAT_1:61;
  (f|X).x = f.x by A2,Th48;
  hence thesis by A3,Def3;
end;
