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
  for Y being set, f being Function holds Y|`f = f|(f"Y)
proof
  let Y be set, f be Function;
A1:  Y|`f c= f|(f"Y) by RELAT_1:188;
   f|(f"Y) c= Y|`f
  proof  let x,y be object;
  assume
A2: [x,y] in f|(f"Y);
  then
A3: x in f"Y by RELAT_1:def 11;
A4: [x,y] in f by A2,RELAT_1:def 11;
  f.x in Y by A3,Def7;
  then y in Y by A4,Th1;
  hence thesis by A4,RELAT_1:def 12;
 end;
 hence thesis by A1;
end;
