 reserve I for non empty set;
 reserve i for Element of I;
 reserve F for Group-Family of I;
 reserve G for Group;
reserve S for Subgroup-Family of F;
reserve f for Homomorphism-Family of G, F;

theorem Th62:
  for D being Subgroup-Family of F
  st (for i being Element of I holds D.i = (F.i)`)
  holds sum D is strict Subgroup of (product F)`
proof
  let D be Subgroup-Family of F;
  assume A1: for i being Element of I holds D.i = (F.i)`;
  sum D is Subgroup of product D & product D is Subgroup of product F;
  then A2: sum D is Subgroup of product F;
  for g being Element of product F st g in sum D holds g in (product F)`
  proof
    let g be Element of product F;
    assume B1: g in sum D;
    1_(product D) = 1_(product F) by GROUP_2:44;
    then consider J being finite Subset of I, b being ManySortedSet of J
    such that
    B2: J = support (g,D) and
    B3: g = ((1_(product F)) +* b) and
    (for j being object for G being Group st j in I \ J & G = D.j
     holds g.j = 1_G) and
    B5: for j being object st j in J holds g.j = b.j
    by B1, GROUP_19:7;
    deffunc F1() = support (g, D);
    defpred P[set]
    means ex FS being FinSequence of the carrier of product F st
    ex ks being FinSequence of INT
    st (len FS = len ks & rng FS c= commutators (product F)
        & ((1_(product F)) +* (b|$1)) = Product (FS |^ ks));
    C1: F1() is finite by B1;
    C2: P[ {} ]
    proof
      1_((product F)`) in (product F)`;
      then 1_(product F) in (product F)` by GROUP_2:44;
      hence thesis by GROUP_5:73;
    end;
    C3: for x,B being set st x in F1() & B c= F1() & P[B]
    holds P[B \/ {x}]
    proof
      let x,B be set;
      assume D1: x in F1();
      assume D2: B c= F1();
      assume D3: P[B];
      per cases;
      suppose x in B;
        then B \/ {x} = B by XBOOLE_1:12,ZFMISC_1:31;
        hence thesis by D3;
      end;
      suppose not (x in B);
        hence thesis by A1,B1,B2,B5,D1,D2,D3,LmHeartOf62;
      end;
    end;
    P[F1()] from FINSET_1:sch 2(C1, C2, C3);
    hence g in (product F)` by B2, B3, GROUP_5:73;
  end;
  hence sum D is strict Subgroup of (product F)` by A2, GROUP_2:58;
end;
