reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;
reserve W,W1,W2 for Subspace of V;
reserve w,w1,w2 for VECTOR of W;
reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:REAL,D:],D;

theorem Th24:
  V1 = D & d1 = 0.V & A = (the addF of V)||V1 & M = (the Mult of V
  ) | [:REAL,V1:] implies RLSStruct (# D,d1,A,M #) is Subspace of V
proof
  assume that
A1: V1 = D and
A2: d1 = 0.V and
A3: A = (the addF of V)||V1 and
A4: M = (the Mult of V) | [:REAL,V1:];
  set W = RLSStruct (# D,d1,A,M #);
A5: for a for x being VECTOR of W holds a * x = (the Mult of V).(a,x)
  proof
    let a;
    let x be VECTOR of W;
     reconsider aa=a as Element of REAL by XREAL_0:def 1;
    thus a * x = (the Mult of V).[aa,x] by A1,A4,FUNCT_1:49
      .= (the Mult of V).(a,x);
  end;
A6: for x,y being VECTOR of W holds x + y = (the addF of V).(x,y)
  proof
    let x,y be VECTOR of W;
    thus x + y = (the addF of V).[x,y] by A1,A3,FUNCT_1:49
      .= (the addF of V).(x,y);
  end;
A7: W is Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
  proof
    set MV = the Mult of V;
    set AV = the addF of V;
    thus W is Abelian
    proof
      let x,y be VECTOR of W;
      reconsider x1 = x, y1 = y as VECTOR of V by A1,TARSKI:def 3;
      thus x + y = x1 + y1 by A6
        .= y1 + x1
        .= y + x by A6;
    end;
    thus W is add-associative
    proof
      let x,y,z be VECTOR of W;
      reconsider x1 = x, y1 = y, z1 = z as VECTOR of V by A1,TARSKI:def 3;
      thus (x + y) + z = AV.(x + y,z1) by A6
        .= (x1 + y1) + z1 by A6
        .= x1 + (y1 + z1) by RLVECT_1:def 3
        .= AV.(x1,y + z) by A6
        .= x + (y + z) by A6;
    end;
    thus W is right_zeroed
    proof
      let x be VECTOR of W;
      reconsider y = x as VECTOR of V by A1,TARSKI:def 3;
      thus x + 0.W = y + 0.V by A2,A6
        .= x;
    end;
    thus W is right_complementable
    proof
      let x be VECTOR of W;
      reconsider x1 = x as VECTOR of V by A1,TARSKI:def 3;
      consider v such that
A8:   x1 + v = 0.V by ALGSTR_0:def 11;
      v = - x1 by A8,RLVECT_1:def 10
        .= (- 1) * x1 by RLVECT_1:16
        .= (- jj) * x by A5;
      then reconsider y = v as VECTOR of W;
      take y;
      thus thesis by A2,A6,A8;
    end;
    thus for a being Real for x,y being VECTOR of W holds a * (x + y) =
    a * x + a * y
    proof
      let a be Real;
      let x,y be VECTOR of W;
      reconsider x1 = x, y1 = y as VECTOR of V by A1,TARSKI:def 3;
      reconsider a as Real;
      a * (x + y) = MV.(a,x + y) by A5
        .= a * (x1 + y1) by A6
        .= a * x1 + a * y1 by RLVECT_1:def 5
        .= AV.(MV.(a,x1),a * y) by A5
        .= AV.(a * x, a * y) by A5
        .= a * x + a * y by A6;
      hence thesis;
    end;
    thus for a,b being Real for x being VECTOR of W holds (a + b) * x =
    a * x + b * x
    proof
      let a,b be Real;
      let x be VECTOR of W;
      reconsider y = x as VECTOR of V by A1,TARSKI:def 3;
      reconsider a,b as Real;
      (a + b) * x = (a + b) * y by A5
        .= a * y + b * y by RLVECT_1:def 6
        .= AV.(MV.(a,y),b * x) by A5
        .= AV.(a * x,b * x) by A5
        .= a * x + b * x by A6;
      hence thesis;
    end;
    thus for a,b being Real for x being VECTOR of W holds (a * b) * x =
    a * (b * x)
    proof
      let a,b be Real;
      let x be VECTOR of W;
      reconsider y = x as VECTOR of V by A1,TARSKI:def 3;
      reconsider a,b as Real;
      (a * b) * x = (a * b) * y by A5
        .= a * (b * y) by RLVECT_1:def 7
        .= MV.(a,b * x) by A5
        .= a * (b * x) by A5;
      hence thesis;
    end;
    let x be VECTOR of W;
    reconsider y = x as VECTOR of V by A1,TARSKI:def 3;
    thus 1 * x = jj * y by A5
      .= x by RLVECT_1:def 8;
  end;
  0.W = 0.V by A2;
  hence thesis by A1,A3,A4,A7,Def2;
end;
