
theorem Th32:
  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 = b * a
  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 as Element of product F;
    reconsider d = b * a as Element of product F;
    A2: dom c = I by Th3;
    A3: dom d = I by Th3;
    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; then
      a.i in F.i by Th5; then
      reconsider ai = a.i as Element of F.i;
      b in product F; then
      b.i in F.i by Th5; then
      reconsider bi = b.i as Element of F.i;
      per cases;
      suppose
        i in support(a,F); then
        A4: not i in support(b,F) by A1,XBOOLE_0:3;
        c.i = ai * bi by GROUP_7:1
           .= ai * 1_F.i by A4,Def1
           .= ai by GROUP_1:def 4
           .= 1_F.i * ai by GROUP_1:def 4
           .= bi * ai by A4,Def1
           .= d.i by GROUP_7:1;
        hence thesis;
      end;
      suppose
        A5: not i in support(a,F);
        c.i = ai * bi by GROUP_7:1
           .= 1_F.i * bi by A5,Def1
           .= bi by GROUP_1:def 4
           .= bi * 1_F.i by GROUP_1:def 4
           .= bi * ai by A5,Def1
           .= d.i by GROUP_7:1;
        hence thesis;
      end;
    end;
    hence thesis by A2,A3,FUNCT_1:2;
  end;
