reserve x,x1,x2,y,y9,y1,y2,z,z1,z2 for object,P,X,X1,X2,Y,Y1,Y2,V,Z for set;
reserve A for set,
  f,g,h for Function;

theorem
 for f being Y-valued Function st x in dom(f|X)
  holds  (f|X)/.x = f/.x
proof let f be Y-valued Function;
 assume
A1: x in dom(f|X);
  then
A2: x in dom f by RELAT_1:57;
 thus (f|X)/.x = (f|X).x by A1,Def6
      .= f.x by A1,FUNCT_1:47
      .= f/.x by A2,Def6;
end;
