
theorem Th19:
  for V being RealLinearSpace, M being non empty Affine Subset of
  V, v being VECTOR of V st v in M holds M - {v} = union {M - {u} where u is
  VECTOR of V : u in M}
proof
  let V be RealLinearSpace;
  let M be non empty Affine Subset of V;
  let v be VECTOR of V;
  assume
A1: v in M;
  for x being object st x in union {M - {u} where u is VECTOR of V : u in M }
  holds x in M - {v}
  proof
    let x be object;
    assume x in union {M - {u} where u is VECTOR of V : u in M};
    then consider N being set such that
A2: x in N and
A3: N in {M - {u} where u is VECTOR of V : u in M} by TARSKI:def 4;
    ex v1 being VECTOR of V st N = M - {v1} & v1 in M by A3;
    hence thesis by A1,A2,Th17;
  end;
  then
A4: union {M - {u} where u is VECTOR of V : u in M} c= M - {v};
  for x being object st x in M - {v} holds x in union {M - {u} where u is
  VECTOR of V : u in M}
  proof
    let x be object;
    assume
A5: x in M - {v};
    M - {v} in {M - {u} where u is VECTOR of V : u in M} by A1;
    hence thesis by A5,TARSKI:def 4;
  end;
  then M - {v} c= union {M - {u} where u is VECTOR of V : u in M};
  hence thesis by A4;
end;
