reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;
reserve z for Complex;
reserve x,y for Complex;
reserve r,s,t for natural Number;

theorem Th31:
  Newton_Coeff s = (1,1) In_Power s
proof
A1: for i st i in dom (Newton_Coeff s) holds (Newton_Coeff s).i = ((1,1)
  In_Power s).i
  proof
    let i;
    assume
A2: i in dom (Newton_Coeff s);
A3: dom (Newton_Coeff s) = Seg len (Newton_Coeff s) by FINSEQ_1:def 3
      .= Seg (s+1) by Def5;
    then i>=1 by A2,FINSEQ_1:1;
    then reconsider m1 = i-1 as Element of NAT by INT_1:5;
    i<=(s+1) by A2,A3,FINSEQ_1:1;
    then s+1-1>=i-1 by XREAL_1:9;
    then reconsider l1 = s-m1 as Element of NAT by INT_1:5;
    dom (Newton_Coeff s) = Seg len ((1,1) In_Power s) by A3,Def4
      .= dom ((1,1) In_Power s) by FINSEQ_1:def 3;
    then ((1,1) In_Power s).i = (s choose m1)*1|^l1*1|^m1 by A2,Def4
      .= (Newton_Coeff s).i by A2,Def5;
    hence thesis;
  end;
  len (Newton_Coeff s) = s+1 by Def5
    .= len ((1,1) In_Power s) by Def4;
  hence thesis by A1,FINSEQ_2:9;
end;
