reserve n,m,k for Nat,
  x,y for set,
  r for Real;
reserve C,D for non empty finite set,
  a for FinSequence of bool D;

theorem
  for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total &
card C = card D holds FinS((Rlor(F,A)) - r,C) = FinS(F-r,D) & Sum (Rlor(F,A)-r,
  C) = Sum (F-r,D)
proof
  let F be PartFunc of D,REAL, B be RearrangmentGen of C;
  assume that
A1: F is total and
A2: card C = card D;
A3: dom F = D by A1,PARTFUN1:def 2;
  then F|D = F by RELAT_1:68;
  then
A4: FinS(F,D), F are_fiberwise_equipotent by A3,RFUNCT_3:def 13;
  Rlor(F,B), FinS(F,D) are_fiberwise_equipotent by A1,A2,Th23;
  then Rlor(F,B), F are_fiberwise_equipotent by A4,CLASSES1:76;
  then
A5: Rlor(F,B)-r, F-r are_fiberwise_equipotent by RFUNCT_3:51;
A6: dom (Rlor(F,B) - r) = dom Rlor (F,B) by VALUED_1:3;
  then (Rlor(F,B) - r)|C = Rlor(F,B) -r by RELAT_1:68;
  then FinS(Rlor(F,B) - r, C), Rlor(F,B) -r are_fiberwise_equipotent
    by A6,RFUNCT_3:def 13; then
A7: FinS(Rlor(F,B) - r, C), F-r are_fiberwise_equipotent by A5,CLASSES1:76;
A8: dom(F-r) = dom F by VALUED_1:3;
  then (F-r)|D = F-r by RELAT_1:68;
  hence FinS(Rlor(F,B) - r,C) = FinS(F-r,D) by A8,A7,RFUNCT_3:def 13;
  hence Sum(Rlor(F,B)-r,C) = Sum FinS(F-r,D) by RFUNCT_3:def 14
    .= Sum (F-r,D) by RFUNCT_3:def 14;
end;
