reserve x,y for set,
  k,n for Nat,
  i for Integer,
  G for Group,
  a,b,c ,d,e for Element of G,
  A,B,C,D for Subset of G,
  H,H1,H2,H3,H4 for Subgroup of G ,
  N1,N2 for normal Subgroup of G,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem
  <* a,b,c *> |^ d = <* a |^ d,b |^ d,c |^ d *>
proof
  thus <* a,b,c *> |^ d = (<* a *> ^ <* b,c *>) |^ d by FINSEQ_1:43
    .= (<* a *> |^ d) ^ (<* b,c *> |^ d) by Th9
    .= <* a *> |^ d ^ <* b |^ d,c |^ d *> by Th12
    .= <* a |^ d *> ^ <* b |^ d,c |^ d *> by Th11
    .= <* a |^ d,b |^ d,c |^ d *> by FINSEQ_1:43;
end;
