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 Th12:
  for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is
  total & card C = card D holds dom Rland(F,A) = C
proof
  let F be PartFunc of D,REAL, A be RearrangmentGen of C;
A1: len CHI(A,C) = len A & len A <> 0 by Th4,RFUNCT_3:def 6;
  assume F is total & card C = card D;
  then
A2: len MIM(FinS(F,D)) = len CHI(A,C) by Th11;
  thus dom Rland(F,A) c= C;
  let x be object;
  assume x in C;
  then reconsider c = x as Element of C;
  len (MIM(FinS(F,D))(#) CHI(A,C)) = min(len MIM(FinS(F,D)), len CHI(A,C))
  & c is_common_for_dom (MIM(FinS(F,D)) (#) CHI(A,C)) by RFUNCT_3:32,def 7;
  hence thesis by A2,A1,RFUNCT_3:28;
end;
