
theorem Th18:
  for V being RealLinearSpace, M being non empty Affine Subset of
  V holds M - M = union {M - {v} where v is VECTOR of V : v in M}
proof
  let V be RealLinearSpace;
  let M be non empty Affine Subset of V;
  for x being object st x in M - M holds x in union {M - {v} where v is
  VECTOR of V : v in M}
  proof
    let x be object;
    assume
A1: x in M - M;
    then reconsider x as Element of V;
    consider u1,v1 being Element of V such that
A2: x = u1 - v1 & u1 in M and
A3: v1 in M by A1;
    v1 in {v1} by TARSKI:def 1;
    then
A4: x in {p - q where p,q is Element of V : p in M & q in {v1}} by A2;
    M - {v1} in {M - {v} where v is VECTOR of V : v in M} by A3;
    hence thesis by A4,TARSKI:def 4;
  end;
  then
A5: M - M c= union {M - {v} where v is VECTOR of V : v in M};
  for x being object st x in union {M - {v} where v is VECTOR of V : v in M}
  holds x in M - M
  proof
    let x be object;
    assume x in union {M - {v} where v is VECTOR of V : v in M};
    then consider N being set such that
A6: x in N and
A7: N in {M - {v} where v is VECTOR of V : v in M} by TARSKI:def 4;
    consider v1 being VECTOR of V such that
A8: N = M - {v1} and
A9: v1 in M by A7;
    consider p1,q1 being Element of V such that
A10: x = p1 - q1 & p1 in M and
A11: q1 in {v1} by A6,A8;
    q1 = v1 by A11,TARSKI:def 1;
    hence thesis by A9,A10;
  end;
  then union {M - {v} where v is VECTOR of V : v in M} c= M - M;
  hence thesis by A5;
end;
