reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem
  for D be non empty set, F be PartFunc of D,REAL, X be set, r be Real,
  Y being finite set st Y = dom(F|X) holds Sum(F-r,X) = Sum(F,X) - r*card(Y)
proof
  let D be non empty set, F be PartFunc of D,REAL, X be set, r be Real;
  set fx = FinS(F,X);
  let Y be finite set;
  reconsider rr=r as Element of REAL by XREAL_0:def 1;
  set dr = card(Y) |-> rr;
  assume
A1: Y = dom(F|X);
  then len fx = card(Y) by Th67;
  then reconsider xf = fx, rd = dr as Element of (card(Y))-tuples_on REAL by
FINSEQ_2:92;
  FinS(F-r,X) = fx - dr by A1,Th73;
  hence Sum(F-r,X) = Sum xf - Sum rd by RVSUM_1:90
    .= Sum(F,X) - r*card(Y) by RVSUM_1:80;
end;
