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
  <* a,b,c *> |^ <* @i1,@i2,@i3 *> = <* a |^ i1,b |^ i2,c |^ i3 *>
proof
A1: len <* @i1 *> = 1 & len<* @i2,@i3 *> = 2 by FINSEQ_1:39,44;
A2: <* a,b,c *> = <* a *> ^ <* b,c *> & <* @i1,@i2,@i3 *> = <* @i1 *> ^ <* @
  i2,@ i3 *> by FINSEQ_1:43;
  len<* a *> = 1 & len<* b,c *> = 2 by FINSEQ_1:39,44;
  hence <* a,b,c *> |^ <* @i1,@i2,@i3 *> = (<* a *> |^ <* @i1 *>) ^ (<* b,c *>
  |^ <* @i2,@i3 *>) by A1,A2,Th19
    .= <* a |^ i1 *> ^ (<* b,c *> |^ <* @i2,@i3 *>) by Th22
    .= <* a |^ i1 *> ^ <* b |^ i2,c |^ i3 *> by Th23
    .= <* a |^ i1,b |^ i2,c |^ i3 *> by FINSEQ_1:43;
end;
