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 Th11:
  for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is
  total & card C = card D holds len MIM FinS(F,D) = len CHI(A,C)
proof
  let F be PartFunc of D,REAL, A be RearrangmentGen of C;
  assume that
A1: F is total and
A2: card C = card D;
  set a = FinS(F,D);
  reconsider a9 = a as finite Function;
A3: dom F = D by A1,PARTFUN1:def 2;
  then reconsider F9 = F as finite Function by FINSET_1:10;
  reconsider da9 = dom a9, dF9 = dom F9 as finite set;
A4: F|D = F by A3,RELAT_1:68;
  D = dom F /\ D by A3
    .= dom(F|D) by RELAT_1:61;
  then F, FinS(F,D) are_fiberwise_equipotent by A4,RFUNCT_3:def 13;
  then
A5: dom a = Seg len a & card da9 = card dF9 by CLASSES1:81,FINSEQ_1:def 3;
  thus len CHI(A,C) = len A by RFUNCT_3:def 6
    .= card C by Th1
    .= len a by A2,A3,A5,FINSEQ_1:57
    .= len MIM(a) by RFINSEQ:def 2;
end;
