
theorem
  for V being Abelian add-associative vector-distributive scalar-distributive
  scalar-associative scalar-unital non empty
  RLSStruct, M,N being Affine Subset of V holds M + N is Affine
proof
  let V be Abelian add-associative vector-distributive scalar-distributive
  scalar-associative scalar-unital non empty RLSStruct;
  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:def 5
      .= (1 - a) * u1 + (1 - a) * v1 + ( a * u2 + a * v2 ) by RLVECT_1:def 5
      .= (1 - a) * u1 + (1 - a) * v1 + a * u2 + a * v2 by RLVECT_1:def 3
      .= (1 - a) * v1 + ((1 - a) * u1 + 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 in M & (1 - a) * v1 + a * v2 in N by A4,A6,Def4;
    hence thesis by A7;
  end;
  hence thesis;
end;
