reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem Th121:
  for R be Relation, X, Y be set st X c= Y holds (R|Y).:X = R.:X
proof
  let R be Relation, X, Y be set such that
A1: X c= Y;
  thus (R|Y).:X c= R.:X by Th53,Th116;
  let y be object;
  assume y in R.:X;
  then consider x1 be object such that
A2: [x1,y] in R and
A3: x1 in X by Def11;
  ex x be object st [x,y] in R|Y & x in X
  proof
    take x1;
    thus [x1,y] in R|Y by A1,A2,A3,Def9;
    thus thesis by A3;
  end;
  hence thesis by Def11;
end;
