
theorem NC7:
  Newton_Coeff 7 = <*1,7,21,35,35,21,7,1*>
  proof
    A1: <*1,6,15,20,15,6,1*> = <*1,6,15,20,15*>^<*6,1*>
      by AOFA_A00:def 5; then
    A2: <*0*>^<*1,6,15,20,15,6,1*> = (<*0*>^<*1,6,15,20,15*>)^<*6,1*>
      by FINSEQ_1:32
    .= <*0*>^(<*1,6,15,20*>^<*15*>)^<*6,1*> by FINSEQ_4:75
    .= (<*0*>^<*1,6,15,20*>)^<*15*>^<*6,1*> by FINSEQ_1:32
    .= (<*0*>^<*1,6,15,20*>)^(<*15*>^<*6,1*>) by FINSEQ_1:32
    .= (<*0*>^<*1,6,15,20*>)^<*15,6,1*> by FINSEQ_1:43
    .= <*0,1,6,15,20*>^<*15,6,1*> by FINSEQ_4:75;
    A3: <*1,6,15,20,15,6,1*>^<*0*> = <*1,6,15,20,15*>^<*6,1,0*>
      by A1,FINSEQ_1:32;
    A4: <*0*>^<*1,6,15,20,15,6,1*> = <*0,1,6,15,20,15,6,1*> &
      <*1,6,15,20,15,6,1*>^<*0*> = <*1,6,15,20,15,6,1,0*>
        by A2,A3,AOFA_A00:def 6;
    Newton_Coeff (6 + 1) = <*0,1,6,15,20,15,6,1*> + <*1,6,15,20,15,6,1,0*>
      by A4,PT,NC6
    .= <*0+1,1+6,6+15,15+20,20+15,15+6,6+1,1+0*> by A8B8;
    hence thesis;
  end;
