reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem FX1:
  for X be set,
      I,f be Function holds
    (f|X)*I = (f*I) | I"X
  proof
    let X be set,
    I,f be Function;
    P1: dom ((f|X)*I) = I"(dom (f|X)) by RELAT_1:147;
    P2:I"( dom (f|X)) = I" ((dom f) /\ X) by RELAT_1:61
    .= I"(dom f ) /\ I"X by FUNCT_1:68
    .= dom (f*I) /\ I"X by RELAT_1:147
    .= dom ((f*I) | I"X) by RELAT_1:61;
    now
      let x be object;
      assume Q1: x in dom ((f|X)*I); then
      x in I"(dom (f|X)) by RELAT_1:147; then
      Q3: x in dom I & I.x in dom (f|X) by FUNCT_1:def 7;
      thus ((f|X)*I ).x = (f|X).(I.x) by FUNCT_1:12,Q1
      .= f.(I.x) by FUNCT_1:47,Q3
      .= (f*I).x by FUNCT_1:13,Q3
      .= ((f*I) | I"X).x by FUNCT_1:47,Q1,P1,P2;
    end;
    hence thesis by P2,FUNCT_1:2,RELAT_1:147;
  end;
