
theorem Th31:
  for V being Abelian add-associative vector-distributive scalar-distributive
  scalar-associative scalar-unital non
  empty RLSStruct, M being Affine Subset of V, v being VECTOR of V holds v + M
  is Affine
proof
  let V be Abelian add-associative vector-distributive scalar-distributive
  scalar-associative scalar-unital non empty RLSStruct;
  let M be Affine Subset of V;
  let v be VECTOR of V;
  for x,y being VECTOR of V, a being Real
    st x in v + M & y in v + M holds (1-a)*x + a*y in v + M
  proof
    let x,y be VECTOR of V;
    let a be Real;
    assume that
A1: x in v + M and
A2: y in v + M;
    consider x9 being Element of V such that
A3: x = v + x9 and
A4: x9 in M by A1;
    consider y9 being Element of V such that
A5: y = v + y9 and
A6: y9 in M by A2;
A7: (1 - a) * x + a * y = (1-a)*v + (1-a)*x9 + a * (v + y9) by A3,A5,
RLVECT_1:def 5
      .= (1-a)*v + (1-a)*x9 + (a*v + a*y9) by RLVECT_1:def 5
      .= (1-a)*v + (1-a)*x9 + a*v + a*y9 by RLVECT_1:def 3
      .= (1-a)*x9 + ((1-a)*v + a*v) + a*y9 by RLVECT_1:def 3
      .= (1-a)*x9 + (1-a+a)*v + a*y9 by RLVECT_1:def 6
      .= (1-a)*x9 + v + a*y9 by RLVECT_1:def 8
      .= v + ((1-a)*x9 + a*y9) by RLVECT_1:def 3;
    (1 - a) * x9 + a * y9 in M by A4,A6,Def4;
    hence thesis by A7;
  end;
  hence thesis;
end;
