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 Th23:
  <* a,b *> |^ <* @i,@j *> = <* a |^ i,b |^ j *>
proof
A1: len <* @i *> = 1 & len<* @j *> = 1 by FINSEQ_1:39;
A2: <* a,b *> = <* a *> ^ <* b *> & <* @i,@j *> = <* @i *> ^ <* @j *> by
FINSEQ_1:def 9;
  len<* a *> = 1 & len<* b *> = 1 by FINSEQ_1:39;
  hence
  <* a,b *> |^ <* @i,@j *> = (<* a *> |^ <* @i *>) ^ (<* b *> |^ <* @j *>
  ) by A1,A2,Th19
    .= <* a |^ i *> ^ (<* b *> |^ <* @j *>) by Th22
    .= <* a |^ i *> ^ <* b |^ j *> by Th22
    .= <* a |^ i,b |^ j *> by FINSEQ_1:def 9;
end;
