
theorem
  for I be non empty set,
      G be Group,
      a be finite-support Function of I,G,
      W be finite Subset of I
  st support a c= W
  holds
  ex aW be finite-support Function of W,G
  st aW = a|W
   & support a = support(aW)
   & Product a = Product(aW)
  proof
    let I be non empty set,
        G be Group,
        a be finite-support Function of I,G,
        W be finite Subset of I;
    assume
    A1: support a c= W;
    A2: for i be object holds i in support a iff i in support(a|W)
    proof
      let i be object;
      hereby
        assume
        A3: i in support a; then
        A4: a.i <> 1_G & i in I by Def2;
        (a|W).i = a.i by A1,A3,FUNCT_1:49;
        hence i in support (a|W) by A1,A3,A4,Def2;
      end;
      assume
      A6: i in support(a|W); then
      A7: (a|W).i <> 1_G & i in W by Def2;
      (a|W).i = a.i by A6,FUNCT_1:49;
      hence i in support a by A7,Def2;
    end;
    support(a|W) is finite; then
    reconsider aW = a|W as finite-support Function of W,G by Def3;
    take aW;
    aW|support(aW) = (a|W) | support(a) by A2,TARSKI:2
                  .= a|support(a) by A1,RELAT_1:74;
    hence thesis by A2,TARSKI:2;
  end;
