reserve x,y for object,
        D,D1,D2 for non empty set,
        i,j,k,m,n for Nat,
        f,g for FinSequence of D*,
        f1 for FinSequence of D1*,
        f2 for FinSequence of D2*;
reserve f for complex-valued Function,
        g,h for complex-valued FinSequence;

theorem Th14:
  k <= m & m <= n implies
  (f,k) +...+ (f,m) + (f,m+1) +...+ (f,n) = (f,k) +...+ (f,n)
proof
  assume A1: k <= m & m <= n;
  defpred P[Nat] means
    (f,k) +...+ (f,m) + (f,m+1) +...+ (f,m+$1) =
       (f,k) +...+ (f,m+$1);
  A2:P[0]
  proof
    m+1 > m+0 by NAT_1:13;
    then (f,m+1) +...+ (f,m+0)=0 by Def1;
    hence thesis;
  end;
  A3:  P[i] implies P[i+1]
  proof
    assume A4:P[i];
    A5:m+1 <= m+1+i by NAT_1:11;
    m <= m+(i+1) by NAT_1:11;
    hence (f,k) +...+ (f,m+(i+1)) = (f,k)+...+(f,m+i)+f.(m+i+1)
                                    by A1,XXREAL_0:2,Th12
      .= (f,k) +...+ (f,m) + ((f,m+1) +...+ (f,m+i) + f.(m+i+1)) by A4
      .= (f,k) +...+ (f,m) + (f,m+1) +...+ (f,m+(i+1)) by Th12,A5;
  end;
  A6:P[i] from NAT_1:sch 2(A2,A3);
  reconsider nm=n-m as Nat by A1,NAT_1:21;
  P[nm] by A6;
  hence thesis;
end;
