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
  F |^ (len F |-> @(0)) = len F |-> 1_G
proof
  defpred P[FinSequence of the carrier of G] means $1 |^ (len $1 |-> @(0)) =
  len $1 |-> 1_G;
A1: for F,a st P[F] holds P[F ^ <* a *>]
  proof
    let F,a;
    set A = F ^ <* a *>;
    assume
A2: P[F];
A3: len<* a *> = 1 by FINSEQ_1:40;
A4: len<* @(0) *> = 1 & len(len F |-> @(0)) = len F by CARD_1:def 7;
A5: len A = len F + len<* a *> by FINSEQ_1:22;
    hence A |^ (len A |-> @(0)) = A |^ ((len F |-> @(0)) ^ <* @(0) *>) by A3,
FINSEQ_2:60
      .= (F |^ (len F |-> @(0))) ^ (<* a *> |^ <* @(0) *>) by A3,A4,Th19
      .= (len F |-> 1_G) ^ <* a |^ 0 *> by A2,Th22
      .= (len F |-> 1_G) ^ <* 1_G *> by GROUP_1:25
      .= (len A |-> 1_G) by A5,A3,FINSEQ_2:60;
  end;
A6: P[<*> the carrier of G]
  proof
    set A = <*> the carrier of G;
    thus A |^ (len A |-> @(0)) = A |^ (0 |-> @(0)) .= A |^ (<*> INT)
      .= {} by Th21
      .= len A |-> 1_G;
  end;
  for F holds P[F] from FINSEQ_2:sch 2(A6,A1);
  hence thesis;
end;
