
theorem Th4:
  for V being RealLinearSpace, M,N being Affine Subset of V holds M
  - N is Affine
proof
  let V be RealLinearSpace;
  let M,N be Affine Subset of V;
  for x,y being VECTOR of V, a being Real
     st x in M - N & y in M - N holds
  (1 - a) * x + a * y in M - N
  proof
    let x,y be VECTOR of V;
    let a be Real;
    assume that
A1: x in M - N and
A2: y in M - N;
    consider u1,v1 being Element of V such that
A3: x = u1 - v1 and
A4: u1 in M & v1 in N by A1;
    consider u2,v2 being Element of V such that
A5: y = u2 - v2 and
A6: u2 in M & v2 in N by A2;
A7: (1 - a) * x + a * y = (1-a)*u1 - (1-a)*v1 + a * (u2 - v2) by A3,A5,
RLVECT_1:34
      .= (1-a)*u1 - (1-a)*v1 + (a*u2 - a*v2) by RLVECT_1:34
      .= (1-a)*u1 + (-(1-a)*v1) + a*u2 - a*v2 by RLVECT_1:def 3
      .= (1-a)*u1 + a*u2 + (-(1-a)*v1) + (-a*v2) by RLVECT_1:def 3
      .= (1-a)*u1 + a*u2 + ((-(1-a)*v1) + (-a*v2)) by RLVECT_1:def 3
      .= (1-a)*u1 + a*u2 - ((1-a)*v1 + a*v2) by RLVECT_1:31;
    (1-a)*u1 + a*u2 in M & (1-a)*v1 + a*v2 in N by A4,A6,RUSUB_4:def 4;
    hence thesis by A7;
  end;
  hence thesis by RUSUB_4:def 4;
end;
