
theorem SumReplace:
  for f be real-valued FinSequence,
      i,j be Nat,
      a, b be Real st
     i in dom f & j in dom f & i <> j holds
    Sum Replace (f,i,j,a,b) = Sum f - f.i - f.j + a + b
  proof
    let f be real-valued FinSequence,
        i,j be Nat,
        a, b be Real;
    assume
A0: i in dom f & j in dom f & i <> j;
A1: j in dom (f +* (i,a)) by A0,FUNCT_7:30;
    Sum Replace (f,i,j,a,b) =
       Sum (f +* (i,a)) - (f +* (i,a)).j + b by A1,SumA
       .= Sum (f +* (i,a)) - f.j + b by FUNCT_7:32,A0
       .= Sum f - f.i + a - f.j + b by A0,SumA
       .= Sum f - f.i + (a - f.j) + b;
    hence thesis;
  end;
