
theorem LMQ20:
  for V be RealLinearSpace, W be Subspace of V holds
  ex QL be LinearOperator of V, VectQuot(V,W)
  st QL is onto & for v be VECTOR of V holds QL.v = v + W
  proof
    let V be RealLinearSpace, W be Subspace of V;
    defpred P[VECTOR of V,object] means $2 = $1 + W;
    A1: for x being Element of the carrier of V
    ex y being Element of the carrier of VectQuot(V,W) st P[x,y]
    proof
      let x be Element of the carrier of V;
      reconsider v = x + W as Point of VectQuot(V,W) by LMQ07;
      take v;
      thus thesis;
    end;
    consider QL being Function of the carrier of V, VectQuot(V,W) such that
    A2: for x being Element of V holds P[x, QL.x] from FUNCT_2:sch 3(A1);
    A3: for v,w be Element of V holds QL.(v+w) = QL.v + QL.w
    proof
      let v,w be Element of V;
      A4: QL.v = v + W by A2;
      A5: QL.w = w + W by A2;
      thus QL.(v+w) = (v+w) + W by A2
                   .= QL.v + QL.w by A4,A5,LMQ11;
    end;
    for v being VECTOR of V, r being Real holds QL.(r*v) = r * QL.v
    proof
      let v be VECTOR of V, r be Real;
      A6: QL.v = v + W by A2;
      thus QL.(r*v) = (r*v) + W by A2
                   .= r * QL.v by A6,LMQ09;
    end; then
    QL is additive homogeneous by A3,LOPBAN_1:def 5; then
    reconsider QL as LinearOperator of V, VectQuot(V,W);
    take QL;
    for v being object st v in the carrier of VectQuot(V,W)
    ex s being object st s in the carrier of V & v = QL.s
    proof
      let v be object;
      assume v in the carrier of VectQuot(V,W); then
      reconsider v1 = v as Point of VectQuot(V,W);
      consider s be VECTOR of V such that
      A7: v1 = s + W by LMQ07;
      take s;
      thus s in the carrier of V;
      thus v = QL.s by A2,A7;
    end;
    hence thesis by A2,FUNCT_2:10;
  end;
