
theorem Th28:
  for F being non empty finite set, A being non empty FinSequence of bool F,
      X being set, B being Reduction of A st
  X is_a_system_of_different_representatives_of B holds
  X is_a_system_of_different_representatives_of A
proof
  let F be non empty finite set, A be non empty FinSequence of bool F, X be
  set, B be Reduction of A such that
A1: X is_a_system_of_different_representatives_of B;
  X is_a_system_of_different_representatives_of A
  proof
    consider f being FinSequence of F such that
A2: f = X and
A3: dom B = dom f and
A4: for i being Element of NAT st i in dom f holds f.i in B.i and
A5: f is one-to-one by A1;
A6: for i being Element of NAT st i in dom f holds f.i in A.i
    proof
      let i be Element of NAT such that
A7:   i in dom f;
A8:   f.i in B.i by A4,A7;
      dom B = dom A by Def6;
      then B.i c= A.i by A3,A7,Def6;
      hence thesis by A8;
    end;
    dom A = dom B by Def6;
    hence thesis by A2,A3,A5,A6;
  end;
  hence thesis;
end;
