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 Th20:
  for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is
  total & card C = card D holds dom Rlor(F,A) = C
proof
  let F be PartFunc of D,REAL, B be RearrangmentGen of C;
  set b = Co_Gen(B);
A1: len CHI(b,C) = len b & len (MIM(FinS(F,D))(#) CHI(b,C)) = min(len MIM(
  FinS(F,D)), len CHI(b,C)) by RFUNCT_3:def 6,def 7;
  assume F is total & card C = card D;
  then
A2: len MIM(FinS(F,D)) = len CHI(b,C) & len b = card D by Th1,Th11;
  thus dom Rlor(F,B) c= C;
  let x be object;
  assume x in C;
  then reconsider c = x as Element of C;
  c is_common_for_dom (MIM(FinS(F,D)) (#) CHI(b,C)) by RFUNCT_3:32;
  hence thesis by A1,A2,RFUNCT_3:28;
end;
