reserve a,b,r for Real;

theorem Th2:
  for V be Abelian add-associative right_zeroed
vector-distributive scalar-distributive scalar-associative scalar-unital
 non empty RLSStruct, V1 be non empty Subset of V, d1 be
Element of V1, A be BinOp of V1, M be Function of [:REAL,V1:],V1 st d1 = 0.V &
A = (the addF of V)|| V1 & M = (the Mult of V)|[:REAL,V1:] holds RLSStruct(# V1
    ,d1,A,M #) is Abelian add-associative right_zeroed vector-distributive
    scalar-distributive scalar-associative scalar-unital
proof
  let V be Abelian add-associative right_zeroed vector-distributive
  scalar-distributive scalar-associative scalar-unital non
empty RLSStruct, V1 be non empty Subset of V, d1 be Element of V1, A be BinOp
  of V1, M be Function of [:REAL,V1:],V1;
  set D = V1;
  assume that
A1: d1 = 0.V and
A2: A = (the addF of V)||V1 and
A3: M = (the Mult of V)|[:REAL,V1:];
  set W = RLSStruct(# D,d1,A,M #);
A4: now
    let a;
     reconsider aa=a as Element of REAL by XREAL_0:def 1;
    let x be VECTOR of W;
    thus a * x = (the Mult of V).[aa,x] by A3,FUNCT_1:49
      .= (the Mult of V).(a,x);
  end;
A5: now
    let x,y be VECTOR of W;
    thus x + y = (the addF of V).[x,y] by A2,FUNCT_1:49
      .= (the addF of V).(x,y);
  end;
  W is Abelian add-associative right_zeroed vector-distributive
  scalar-distributive scalar-associative scalar-unital
  proof
    set Mv = the Mult of V;
    set Av = the addF of V;
    hereby
      let x,y be VECTOR of W;
      reconsider x1 = x, y1 = y as VECTOR of V by TARSKI:def 3;
      thus x + y = x1 + y1 by A5
        .= y1 + x1
        .= y + x by A5;
    end;
    now
      let x,y,z be VECTOR of W;
      reconsider x1 = x, y1 = y, z1 = z as VECTOR of V by TARSKI:def 3;
      thus (x + y) + z = Av.(x + y,z1) by A5
        .= (x1 + y1) + z1 by A5
        .= x1 + (y1 + z1) by RLVECT_1:def 3
        .= Av.(x1,y + z) by A5
        .= x + (y + z) by A5;
    end;
    hence W is add-associative;
    now
      let x be VECTOR of W;
      reconsider y = x as VECTOR of V by TARSKI:def 3;
      thus x + 0.W = y + 0.V by A1,A5
        .= x by RLVECT_1:def 4;
    end;
    hence W is right_zeroed;
    hereby
      let a1 be Real;
      let x,y be VECTOR of W;
      reconsider a=a1 as Real;
      reconsider x1 = x, y1 = y as VECTOR of V by TARSKI:def 3;
      a * (x + y) = Mv.(a,x + y) by A4
        .= a * (x1 + y1) by A5
        .= a * x1 + a * y1 by RLVECT_1:def 5
        .= Av.(Mv.(a,x1),a * y) by A4
        .= Av.(a * x, a * y) by A4
        .= a * x + a * y by A5;
      hence a1 * (x + y)= a1 * x + a1 * y;
    end;
    hereby
      let a1,b1 be Real;
      let x be VECTOR of W;
      reconsider a=a1,b=b1 as Real;
      reconsider y = x as VECTOR of V by TARSKI:def 3;
      (a + b) * x = (a + b) * y by A4
        .= a * y + b * y by RLVECT_1:def 6
        .= Av.(Mv.(a,y),b * x) by A4
        .= Av.(a * x,b * x) by A4
        .= a * x + b * x by A5;
      hence (a1 + b1) * x= a1 * x + b1 * x;
    end;
    hereby
      let a1,b1 be Real;
      let x be VECTOR of W;
      reconsider a=a1,b=b1 as Real;
      reconsider y = x as VECTOR of V by TARSKI:def 3;
      (a * b) * x = (a * b) * y by A4
        .= a * (b * y) by RLVECT_1:def 7
        .= Mv.(a,b * x) by A4
        .= a * (b * x) by A4;
      hence (a1 * b1) * x= a1 * (b1 * x);
    end;
    let x be VECTOR of W;
    reconsider y = x as VECTOR of V by TARSKI:def 3;
    thus 1 * x = 1 * y by A4
      .= x by RLVECT_1:def 8;
  end;
  hence thesis;
end;
