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;
reserve e,u for object,
  A for Subset of X;

theorem Th107:
 x in X & x in dom f implies f.x in f.:X
  proof assume that
A1: x in X and
A2: x in dom f;
   x in X /\ dom f by A1,A2,XBOOLE_0:def 4;
   then x in dom(f|X) by RELAT_1:61;
   then
A3:  (f|X).x in rng(f|X) by Def3;
    (f|X).x = f.x by A1,Th48;
   hence f.x in f.:X by A3,RELAT_1:115;
  end;
