
theorem Th2:
  for I be set, F be Function st I = dom F
   & for i be object st i in I holds F.i is Group
  holds F is Group-Family of I
  proof
    let I be set;
    let F be Function;
    assume
    A1: I = dom F;
    assume
    A2: for i be object st i in I holds F.i is Group;
    reconsider F as ManySortedSet of I by A1,PARTFUN1:def 2,RELAT_1:def 18;
    now
      let y be set;
      assume y in rng F; then
      consider i be object such that
      A3: i in dom F & y = F.i by FUNCT_1:def 3;
      thus y is non empty multMagma by A2,A3;
    end; then
    reconsider F as multMagma-Family of I by GROUP_7:def 1;
    A4: for i being set st i in I ex Fi being
        Group-like non empty multMagma st Fi = F.i
    proof
      let i be set;
      assume i in I; then
      reconsider Fi = F.i as Group-like non empty multMagma by A2;
      take Fi;
      thus thesis;
    end;
    for i being set st i in I ex Fi being
        associative non empty multMagma st Fi = F.i
    proof
      let i be set;
      assume i in I; then
      reconsider Fi = F.i as associative non empty multMagma by A2;
      take Fi;
      thus thesis;
    end;
    hence thesis by A4,GROUP_7:def 3,def 4;
  end;
