
theorem
  for X be set for x be Subset of X for q be FinSequence of BOOLEAN
holds (q.1 = TRUE implies MergeSequence(<*x*>,q).1 = x) & (q.1 = FALSE implies
  MergeSequence(<*x*>,q).1 = X\x)
proof
  let X be set;
  let x be Subset of X;
  let q be FinSequence of BOOLEAN;
  thus q.1 = TRUE implies MergeSequence(<*x*>,q).1 = x
  proof
    assume q.1 = TRUE;
    hence MergeSequence(<*x*>,q).1 = <*x*>.1 by Th2
      .= x;
  end;
  1 in Seg 1 by FINSEQ_1:1;
  then
A1: 1 in dom <*x*> by FINSEQ_1:38;
  assume q.1 = FALSE;
  hence MergeSequence(<*x*>,q).1 = X\<*x*>.1 by A1,Th3
    .= X\x;
end;
