reserve i, I for set,
  f, g, h for Function,
  s for ManySortedSet of I;

theorem Th6:
  for F being Group-like multMagma-Family of I,
      G being Group-like non empty multMagma st
    i in I & G = F.i & f = 1_product F holds f.i = 1_G
proof
  let F be Group-like multMagma-Family of I,
      G be Group-like non empty multMagma such that
A1: i in I and
A2: G = F.i and
A3: f = 1_product F;
  set GP = product F;
  f in the carrier of GP by A3;
  then
A4: f in product Carrier F by Def2;
  then reconsider e = f.i as Element of G by A1,A2,Lm1;
  now
    let h be Element of G;
    defpred P[object,object] means
($1 = i implies $2 = h) & ($1 <> i implies ex H
    being Group-like non empty multMagma st H = F.$1 & $2 = 1_H);
A5: for j being object st j in I ex k being object st P[j,k]
    proof
      let j be object such that
A6:   j in I;
      per cases;
      suppose
        j = i;
        hence thesis;
      end;
      suppose
A7:     j <> i;
        consider Fj being Group-like non empty multMagma such that
A8:     Fj = F.j by A6,Def3;
        take 1_Fj;
        thus j = i implies 1_Fj = h by A7;
        thus thesis by A8;
      end;
    end;
    consider g being ManySortedSet of I such that
A9: for j being object st j in I holds P[j,g.j] from PBOOLE:sch 3(A5);
A10: dom g = I by PARTFUN1:def 2;
A11: now
      let j be object;
      assume
A12:  j in dom g;
      then
A13:  ex R being 1-sorted st R = F.j & (Carrier F).j = the carrier of R
      by PRALG_1:def 15;
      per cases;
      suppose
A14:    i = j;
        then g.j = h by A9,A12;
        hence g.j in (Carrier F).j by A2,A13,A14;
      end;
      suppose
        j <> i;
        then ex H being Group-like non empty multMagma st ( H = F.j)&( g.j =
        1_H) by A9,A12;
        hence g.j in (Carrier F).j by A13;
      end;
    end;
    dom Carrier F = I by PARTFUN1:def 2;
    then
A15: g in product Carrier F by A10,A11,CARD_3:9;
    then reconsider g1 = g as Element of GP by Def2;
A16: g1 * 1_product F = g1 by GROUP_1:def 4;
A17: g.i = h by A1,A9;
A18: g.i = h by A1,A9;
    ex Fi being non empty multMagma, m being Function st Fi = F.i & m =
    (the multF of GP).(g,f) & m.i = (the multF of Fi).(g.i,f.i) by A1,A4,A15
,Def2;
    hence h * e = h by A2,A3,A16,A18;
A19: 1_product F * g1 = g1 by GROUP_1:def 4;
    ex Fi being non empty multMagma, m being Function st Fi = F.i & m =
    (the multF of GP).(f,g) & m.i = (the multF of Fi).(f.i,g.i) by A1,A4,A15
,Def2;
    hence e * h = h by A2,A3,A19,A17;
  end;
  hence thesis by GROUP_1:4;
end;
