reserve i, j, m, n for Nat,
  z, B0 for set,
  f, x0 for real-valued FinSequence;

theorem
  B0 is R-orthogonal iff for x,y being real-valued FinSequence st x in
  B0 & y in B0 & x <> y holds x,y are_orthogonal
proof
  thus B0 is R-orthogonal implies for x,y being real-valued FinSequence st x
  in B0 & y in B0 & x <> y holds x,y are_orthogonal;
  assume
A1: for x,y being real-valued FinSequence st x in B0 & y in B0 & x <> y
  holds x,y are_orthogonal;
  let x,y be real-valued FinSequence;
  assume that
A2: x in B0 and
A3: y in B0 and
A4: x<>y;
  x,y are_orthogonal by A1,A2,A3,A4;
  hence |( x,y )|=0;
end;
