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 Th20:
  for f be finite complex-valued Function holds
    f.n + (f,n+1) +... = (f,n) +...
proof
  let f be finite complex-valued Function;
  {n} c= NAT by ORDINAL1:def 12;
  then reconsider D=(dom f /\NAT)\/{n} as finite non empty Subset of NAT
    by XBOOLE_1:8;
  reconsider m=max D as Nat by TARSKI:1;
  A1:for i st i in dom f holds i <= m
  proof
    let i;
    assume A2:i in dom f;
    i in NAT by ORDINAL1:def 12;
    then i in dom f /\NAT by A2,XBOOLE_0:def 4;
    then i in D by XBOOLE_0:def 3;
    hence thesis by XXREAL_2:def 8;
  end;
  then A3:(f,n+1)+... = (f,n+1)+...+(f,m) by Def2;
  A4:(f,n)+... = (f,n)+...+(f,m) by Def2,A1;
  n in {n} by TARSKI:def 1;
  then n in D by XBOOLE_0:def 3;
  then n <=m by XXREAL_2:def 8;
  hence thesis by Th13,A3,A4;
end;
