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 Th22:
  <* a *> |^ <* @i *> = <* a |^ i *>
proof
A1: len<* a *> = 1 by FINSEQ_1:39;
A2: now
    let k;
A3: @i = <* @i *>/.1 by FINSEQ_4:16;
A4: dom <* a *> = Seg len <* a *> by FINSEQ_1:def 3;
    assume k in dom <* a *>;
    then
A5: k = 1 by A1,A4,FINSEQ_1:2,TARSKI:def 1;
    hence <* a |^ i *>.k = a |^ i
      .= (<* a *>/.k) |^ @(<* @i *>/.k) by A3,A5,FINSEQ_4:16;
  end;
  len<* a |^ i*> = 1 by FINSEQ_1:39;
  hence thesis by A1,A2,Def3;
end;
