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 Th14:
  for c be Element of C, F be PartFunc of D,REAL, A be
RearrangmentGen of C st F is total & card C = card D holds (c in A.1 implies (
  Rland(F,A)).c = FinS(F,D).1) & for n st 1<=n & n<len A & c in A.(n+1) \ A.n
  holds Rland(F,A).c = FinS(F,D).(n+1)
proof
  let c be Element of C, F be PartFunc of D,REAL, B be RearrangmentGen of C;
  set fd = FinS(F,D), mf = MIM(fd), h = CHI(B,C);
A1: Rland(F,B).c = Sum((mf(#)h)#c) by RFUNCT_3:32,33;
A2: len h = len B & len mf = len fd by RFINSEQ:def 2,RFUNCT_3:def 6;
  assume
A3: F is total & card C = card D;
  then
A4: len mf = len h by Th11;
  thus c in B.1 implies Rland(F,B).c = FinS(F,D).1
  proof
    assume c in B.1;
    hence Rland(F,B).c = Sum mf by A3,A1,Th13
      .= FinS(F,D).1 by A4,A2,Th4,RFINSEQ:16;
  end;
  let n;
  set mfn = MIM(FinS(F,D)/^n);
  assume that
A5: 1<=n and
A6: n<len B and
A7: c in B.(n+1) \ B.n;
  (mf(#)h)#c = (n |-> (0 qua Real)) ^ mfn by A3,A5,A6,A7,Th13;
  hence Rland(F,B).c = Sum(n |-> In(0,REAL)) + (Sum mfn)
     by A1,RVSUM_1:75
    .= 0 + Sum mfn by RVSUM_1:81
    .= FinS(F,D).(n+1) by A4,A2,A6,RFINSEQ:17;
end;
