 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 ThCanSubgrFam:
  for G being Group
  for S being finite Subset of Subgroups G
  for n being Nat st n = card S
  holds canFS S is Subgroup-Family of Seg n, G
proof
  let G be Group;
  let S be finite Subset of Subgroups G;
  let n be Nat;
  assume A1: n = card S;
  len (canFS S) = n by A1, FINSEQ_1:93;
  then A2: dom (canFS S) = Seg n by FINSEQ_1:def 3;
  for y being object st y in rng (canFS S) holds y is Subgroup of G
  proof
    let y be object;
    assume y in rng (canFS S);
    then y in S by FINSEQ_1:def 4, TARSKI:def 3;
    hence thesis by GROUP_3:def 1;
  end;
  then canFS S is Group-yielding;
  then A5: canFS S is Group-Family of Seg n by A1, ThCanFSIsMss;
  for i being object st i in (Seg n) holds (canFS S).i is Subgroup of G
  proof
    let i be object;
    assume i in Seg n;
    then (canFS S).i in rng (canFS S) by A2, FUNCT_1:3;
    then (canFS S).i in S by FINSEQ_1:def 4, TARSKI:def 3;
    hence thesis by GROUP_3:def 1;
  end;
  hence thesis by A5, GROUP_20:def 1;
end;
