
theorem NC6:
  Newton_Coeff 6 = <*1,6,15,20,15,6,1*>
  proof
    A1: <*1,5,10,10,5,1*> = (<*1,5,10,10,5*>^<*1*>) by AOFA_A00:def 4;  then
    A2: <*0*>^<*1,5,10,10,5,1*> = (<*0*>^<*1,5,10,10,5*>)^<*1*> by FINSEQ_1:32
    .= (<*0*>^(<*1,5,10,10*>^<*5*>))^<*1*> by FINSEQ_4:75
    .= (<*0*>^<*1,5,10,10*>)^<*5*>^<*1*> by FINSEQ_1:32
    .= (<*0*>^<*1,5,10,10*>)^(<*5*>^<*1*>) by FINSEQ_1:32
    .= <*0,1,5,10,10*>^<*5,1*> by FINSEQ_4:75;
    A3: <*1,5,10,10,5,1*>^<*0*> = <*1,5,10,10,5*>^<*1,0*> by A1,FINSEQ_1:32;
    A4: <*0*>^<*1,5,10,10,5,1*> = <*0,1,5,10,10,5,1*> &
      <*1,5,10,10,5,1*>^<*0*> = <*1,5,10,10,5,1,0*> by A2,A3,AOFA_A00:def 5;
    Newton_Coeff (5 + 1) = <*0,1,5,10,10,5,1*> + <*1,5,10,10,5,1,0*>
      by A4,PT,NC5
    .= <*0+1,1+5,5+10,10+10,10+5,5+1,1+0*> by A7B7;
    hence thesis;
  end;
