reserve x,y for set,
  n,m for Nat,
  r,s for Real;

theorem
  for R be real-valued FinSequence, n be Nat st n<len R
  holds Sum MIM(R/^n) = R.(n+1)
proof
  let R be real-valued FinSequence,n;
  assume
A1: n<len R;
  then
A2: len(R/^n) = len R - n by Def1;
  n+1<=len R by A1,NAT_1:13;
  then 1<=len R - n by XREAL_1:19;
  then
A3: 1 in dom(R/^n) by A2,FINSEQ_3:25;
  len(R/^n) <> 0 by A1,A2;
  hence Sum MIM(R/^n) = (R/^n).1 by Th16
    .= R.(n+1) by A1,A3,Def1;
end;
