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 Th12:
  <* a,b *> |^ c = <* a |^ c,b |^ c *>
proof
  thus <* a,b *> |^ c = (<* a *> ^ <* b *>) |^ c by FINSEQ_1:def 9
    .= (<* a *> |^ c) ^ (<* b *> |^ c) by Th9
    .= <* a |^ c *> ^ (<* b *> |^ c) by Th11
    .= <* a |^ c *> ^ <* b |^ c *> by Th11
    .= <* a |^ c, b |^ c *> by FINSEQ_1:def 9;
end;
