
theorem
  for I be non empty set,
      F be Group-Family of I,
      a,b be Element of product F
  st support(a,F) misses support(b,F)
  holds a +* (b|support(b,F)) = a * b
  proof
    let I be non empty set,
        F be Group-Family of I,
        a,b be Element of product F;
    assume
    A1: support(a,F) misses support(b,F);
    reconsider c = a +* (b|support(b,F)) as Function;
    reconsider d = a * b as Element of product F;
    A2: dom a = I by Th3;
    A3: dom b = I by Th3;
    A5: dom c = dom a \/ dom(b|support(b,F)) by FUNCT_4:def 1
             .= I by A2,A3,RELAT_1:60,XBOOLE_1:12;
    A6: dom d = I by Th3;
    A8: dom(b|support(b,F)) = support(b,F) by A3,RELAT_1:62;
    for i be object st i in I holds c.i = d.i
    proof
      let i be object;
      assume i in I; then
      reconsider i as Element of I;
      a in product F & b in product F; then
      a.i in F.i & b.i in F.i by Th5; then
      reconsider ai = a.i, bi = b.i as Element of F.i;
      per cases;
      suppose
        A9: not i in support(b,F);
        c.i = a.i by A8,A9,FUNCT_4:11
           .= ai * 1_F.i by GROUP_1:def 4
           .= ai * bi by A9,Def1
           .= d.i by GROUP_7:1;
        hence thesis;
      end;
      suppose
        A11: i in support(b,F); then
        A12: not i in support(a,F) by A1,XBOOLE_0:3;
        c.i = (b|support(b,F)).i by A8,A11,FUNCT_4:13
           .= bi by A11,FUNCT_1:49
           .= 1_F.i * bi by GROUP_1:def 4
           .= ai * bi by A12,Def1
           .= d.i by GROUP_7:1;
        hence thesis;
      end;
    end;
    hence thesis by A5,A6,FUNCT_1:2;
  end;
