
theorem NC5:
  Newton_Coeff 5 = <*1,5,10,10,5,1*>
  proof
    <*0*>^<*1,4,6,4,1*> = <*0*>^(<*1,4,6,4*>^<*1*>) by FINSEQ_4:75
    .= (<*0*>^<*1,4,6,4*>)^<*1*> by FINSEQ_1:32
    .= <*0,1,4,6,4*>^<*1*> by FINSEQ_4:75; then
    A1: <*0*>^<*1,4,6,4,1*> = <*0,1,4,6,4,1*> &
      <*1,4,6,4,1*>^<*0*> = <*1,4,6,4,1,0*> by AOFA_A00:def 4;
    Newton_Coeff (4 + 1) = <*0,1,4,6,4,1*> + <*1,4,6,4,1,0*> by A1,PT,NC4
    .= <*0+1,1+4,4+6,6+4,4+1,1+0*> by A6B6;
    hence thesis;
  end;
