reserve a,b for Complex;
reserve V,X,Y for ComplexLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve z,z1,z2 for Complex;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Subspace of V;
reserve x for set;
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 [:COMPLEX,D:],D;

theorem Th43:
  V1 = D & d1 = 0.V & A = (the addF of V)||V1 &
  M = (the Mult of V) | [:COMPLEX,V1:] implies
  CLSStruct (# 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) | [:COMPLEX,V1:];
  set W = CLSStruct (# D,d1,A,M #);
A5: for x,y being VECTOR of W holds x + y = (the addF of V).[x,y] by A1,A3,
FUNCT_1:49;
A6: 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 for x,y being VECTOR of W holds x + y = y + x
    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 A5
        .= y1 + x1
        .= y + x by A5;
    end;
    thus for x,y,z being VECTOR of W holds (x + y) + z = x + (y + z)
    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 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;
    thus for x being VECTOR of W holds x + 0.W = x
    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,A5
        .= x by RLVECT_1:4;
    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
A7:   x1 + v = 0.V by ALGSTR_0:def 11;
      reconsider mj=-1r as Element of COMPLEX by XCMPLX_0:def 2;
      v = - x1 by A7,RLVECT_1:def 10
        .= (- 1r) * x1 by Th3
        .= MV.[mj,x]
        .= (- 1r) * x by A1,A4,FUNCT_1:49;
      then reconsider y = v as VECTOR of W;
      take y;
      thus thesis by A2,A5,A7;
    end;
    thus for z for x,y being VECTOR of W holds z * (x + y) = z * x + z * y
    proof
      let z;
      let x,y be VECTOR of W;
      reconsider x1 = x, y1 = y as VECTOR of V by A1,TARSKI:def 3;
A8:   z*x1 = z*x & z*y1 = z*y by A1,A4,Lm4;
      thus z * (x + y) = z * (x1 + y1) by A1,A4,A5,Lm4
        .= z * x1 + z * y1 by Def2
        .= z * x + z * y by A5,A8;
    end;
    thus for z1,z2 for x being VECTOR of W holds (z1 + z2) * x = z1*x + z2*x
    proof
      let z1,z2;
      let x be VECTOR of W;
      reconsider y = x as VECTOR of V by A1,TARSKI:def 3;
A9:   z1*y = z1*x & z2*y = z2*x by A1,A4,Lm4;
      thus (z1 + z2) * x = (z1 + z2) * y by A1,A4,Lm4
        .= z1 * y + z2 * y by Def3
        .= z1 * x + z2 * x by A5,A9;
    end;
    thus for z1,z2 for x being VECTOR of W holds (z1 * z2) * x = z1 * (z2 * x)
    proof
      let z1,z2;
      let x be VECTOR of W;
      reconsider y = x as VECTOR of V by A1,TARSKI:def 3;
A10:   z2*y = z2*x by A1,A4,Lm4;
      thus (z1 * z2) * x = (z1 * z2) * y by A1,A4,Lm4
        .= z1 * (z2 * y) by Def4
        .= z1 * (z2 * x) by A1,A4,A10,Lm4;
    end;
    let x be VECTOR of W;
    reconsider y = x as VECTOR of V by A1,TARSKI:def 3;
    thus 1r * x = 1r * y by A1,A4,Lm4
      .= x by Def5;
  end;
  0.W = 0.V by A2;
  hence thesis by A1,A3,A4,A6,Def8;
end;
