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 D = card C holds Rland(F,A) , F are_fiberwise_equipotent & Rlor (F,A) , F
  are_fiberwise_equipotent & rng Rland(F,A) = rng F & rng Rlor(F,A) = rng F
proof
  let F be PartFunc of D,REAL, A be RearrangmentGen of C;
  assume that
A1: F is total and
A2: card D = card C;
A3: dom F = D by A1,PARTFUN1:def 2;
  dom(F|D) = dom F /\ D by RELAT_1:61
    .= D by A3;
  then
A4: FinS(F,D), F|D are_fiberwise_equipotent by RFUNCT_3:def 13;
  Rlor(F,A), FinS(F,D) are_fiberwise_equipotent by A1,A2,Th23;
  then
A5: Rlor (F,A), F|D are_fiberwise_equipotent by A4,CLASSES1:76;
  Rland(F,A), FinS(F,D) are_fiberwise_equipotent by A1,A2,Th16;
  then Rland(F,A), F|D are_fiberwise_equipotent by A4,CLASSES1:76;
  hence Rland(F,A), F are_fiberwise_equipotent & Rlor(F,A), F
  are_fiberwise_equipotent by A3,A5,RELAT_1:68;
  hence thesis by CLASSES1:75;
end;
