
theorem Th19:
  for F being finite set, A being FinSequence of bool F,
      i being Element of NAT,
      C being Reduction of A, i holds C is Reduction of A
proof
  let F be finite set, A be FinSequence of bool F, i be Element of NAT, C be
  Reduction of A, i;
A1: dom C = dom A by Def5;
  for j being Element of NAT st j in dom C holds C.j c= A.j
  proof
    let j be Element of NAT;
    assume
A2: j in dom C;
    per cases;
    suppose
      j = i;
      hence thesis by Def5;
    end;
    suppose
      j <> i;
      hence thesis by A1,A2,Def5;
    end;
  end;
  hence thesis by A1,Def6;
end;
