
theorem
  Newton_Coeff 8 = <*1,8,28,56,70,56,28,8,1*>
  proof
    Newton_Coeff (7 + 1) = <*0*>^<*1,7,21,35,35,21,7,1*> +
      <*1,7,21,35,35,21,7,1*>^<*0*> by PT,NC7
    .= <*0*>^(<*1,7,21,35,35*>^<*21,7,1*>) + <*1,7,21,35,35,21,7,1*>^<*0*>
      by AOFA_A00:def 6
    .= <*0*>^<*1,7,21,35,35*>^<*21,7,1*> + <*1,7,21,35,35,21,7,1*>^<*0*>
      by FINSEQ_1:32
    .= <*0*>^(<*1,7,21,35*>^<*35*>)^<*21,7,1*> + <*1,7,21,35,35,21,7,1*>^<*0*>
      by FINSEQ_4:75
    .= <*0*>^<*1,7,21,35*>^<*35*>^<*21,7,1*> + <*1,7,21,35,35,21,7,1*>^<*0*>
      by FINSEQ_1:32
    .= <*0,1,7,21,35*>^<*35*>^<*21,7,1*> + <*1,7,21,35,35,21,7,1*>^<*0*>
      by FINSEQ_4:75
    .= <*0,1,7,21,35*>^(<*35*>^<*21,7,1*>) + <*1,7,21,35,35,21,7,1*>^<*0*>
      by FINSEQ_1:32
    .= <*0,1,7,21,35*>^(<*35,21,7,1*>) + <*1,7,21,35,35,21,7,1*>^<*0*>
      by FINSEQ_4:74
    .= <*0,1,7,21,35*>^(<*35,21,7*>^<*1*>) + <*1,7,21,35,35,21,7,1*>^<*0*>
      by FINSEQ_4:74
    .= <*0,1,7,21,35*>^<*35,21,7*>^<*1*> + <*1,7,21,35,35,21,7,1*>^<*0*>
      by FINSEQ_1:32
    .= <*0,1,7,21,35,35,21,7*>^<*1*> + <*1,7,21,35,35,21,7,1*>^<*0*>
      by AOFA_A00:def 6
    .= (<*0,1,7,21,35,35,21,7*> + <*1,7,21,35,35,21,7,1*>)^<*0+1*>
      by FINSEQ_9:40
    .= <*0+1,1+7,7+21,21+35,35+35,35+21,21+7,7+1*>^<*0+1*> by A8B8;
    hence thesis;
  end;
