reserve x,y,X,Y for set,
  k,l,n for Nat,
  i,i1,i2,i3,j for Integer,
  G for Group,
  a,b,c,d for Element of G,
  A,B,C for Subset of G,
  H,H1,H2, H3 for Subgroup of G,
  h for Element of H,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem Th20:
  rng F c= carr H implies Product(F |^ I) in H
proof
  assume
A1: rng F c= carr H;
  rng(F |^ I) c= carr H
  proof
    let x be object;
    assume x in rng(F |^ I);
    then consider y being object such that
A2: y in dom(F |^ I) and
A3: x = (F |^ I).y by FUNCT_1:def 3;
    reconsider y as Element of NAT by A2;
    len(F |^ I) = len F by Def3;
    then
A4: y in dom F by A2,FINSEQ_3:29;
    then F.y in rng F & F.y = F/.y by FUNCT_1:def 3,PARTFUN1:def 6;
    then
A5: F/.y in H by A1,STRUCT_0:def 5;
    x = (F/.y) |^ @(I/.y) by A3,A4,Def3;
    then x in H by A5,Th4;
    hence thesis by STRUCT_0:def 5;
  end;
  hence thesis by Th18;
end;
