
theorem
  for V,W be RealNormSpace,
      L be Lipschitzian LinearOperator of V,W
  holds
  ex QL be Lipschitzian LinearOperator of NVectQuot(V,Ker L), Im L,
     PQL be Point of R_NormSpace_of_BoundedLinearOperators(NVectQuot(V,Ker L),
     Im L),
     PL be Point of R_NormSpace_of_BoundedLinearOperators(V,W)
  st QL is onto & QL is one-to-one
   & L = PL & QL = PQL & ||.PL.|| = ||.PQL.||
   & for z be Point of NVectQuot(V,Ker L), v be VECTOR of V
     st z = v + Ker L holds QL.z = L.v
  proof
    let V,W be RealNormSpace,
        L be Lipschitzian LinearOperator of V,W;
    A1: the carrier of Ker L = L"{0.W} & L"{0.W} is closed
    by KERCL01,KLXY1,LCL1,LOPBAN_7:6;
    reconsider V1 = V as RealLinearSpace;
    reconsider W1 = W as RealLinearSpace;
    reconsider L1 = L as LinearOperator of V1,W1;
    A2: the RLSStruct of NVectQuot(V,Ker L) = VectQuot (V,Ker L)
      & the normF of NVectQuot(V,Ker L) = NormCoset(V,Ker L) by A1,defQuotN;
    A3: the carrier of VectQuot(V,Ker L) = CosetSet(V,Ker L) by LMQ06;
    consider QL1 be LinearOperator of VectQuot(V1,Ker L1), Im L1 such that
    A4: QL1 is isomorphism
      & for z be Point of VectQuot(V1,Ker L1), v be VECTOR of V1
      st z = v + Ker L1 holds QL1.z = L1.v by LMQ21;
    reconsider KL1 = Ker L1 as Subspace of V;
    A5: the RLSStruct of NVectQuot(V,Ker L) = VectQuot (V,KL1) by A1,defQuotN;
    reconsider QL = QL1 as Function of NVectQuot(V,Ker L),Im L by A5;
    A6: for v,w be Element of NVectQuot(V,Ker L) holds QL.(v+w) = QL.v + QL.w
    proof
      let v,w be Element of NVectQuot(V,Ker L);
      reconsider v1 = v, w1 = w as Element of VectQuot(V1,Ker L1) by A5;
      thus QL.(v+w) = QL1.(v1+w1) by A5
                   .= QL1.v1 + QL1.w1 by VECTSP_1:def 20
                   .= QL.v + QL.w;
    end;
    for v being VECTOR of NVectQuot(V,Ker L), r being Real
    holds QL.(r * v) = r * QL.v
    proof
      let v be VECTOR of NVectQuot(V,Ker L), r be Real;
      reconsider v1 = v as Element of VectQuot(V1,Ker L1) by A5;
      thus QL.(r*v) = QL1.(r*v1) by A5
                   .= r * QL1.v1 by LOPBAN_1:def 5
                   .= r * QL.v;
    end; then
    QL is additive homogeneous by A6,LOPBAN_1:def 5; then
    reconsider QL as LinearOperator of NVectQuot(V,Ker L), Im L;
    A7: R_NormSpace_of_BoundedLinearOperators(V,W)
          = NORMSTR (# BoundedLinearOperators(V,W),
                       Zero_(BoundedLinearOperators(V,W),
                       R_VectorSpace_of_LinearOperators(V,W)),
                       Add_(BoundedLinearOperators(V,W),
                       R_VectorSpace_of_LinearOperators(V,W)),
                       Mult_(BoundedLinearOperators(V,W),
                       R_VectorSpace_of_LinearOperators(V,W)),
                       BoundedLinearOperatorsNorm(V,W) #) by LOPBAN_1:def 14;
    reconsider PL = L as Point of R_NormSpace_of_BoundedLinearOperators(V,W)
    by A7,LOPBAN_1:def 9;
    A8: for v be Point of NVectQuot(V,Ker L)
    holds ||.QL.v.|| <= ||.PL.|| * ||.v.||
    proof
      let v be Point of NVectQuot(V,Ker L);
      reconsider v1 = v as Element of VectQuot(V,Ker L) by A5;
      consider vv1 be Point of V such that
      A9: v1 = vv1 + Ker L by LMQ07;
      A10: ||.v.|| = (NormCoset(V,Ker L)).v1 by A1,defQuotN
                  .= lower_bound NormVSets(V,Ker L,vv1) by A3,A9,DeNorm;
      per cases;
      suppose
        ||.PL.|| = 0; then
        PL = 0.R_NormSpace_of_BoundedLinearOperators(V,W) by NORMSP_0:def 5;
        then
        A11: (the carrier of V) --> 0.W = L by LOPBAN_1:31;
        QL.v = L.vv1 by A4,A9
            .= 0.W by A11,FUNCOP_1:7; then
        ||.QL.v.|| = ||.0.W.|| by SUBTH0;
        hence ||.QL.v.|| <= ||.PL.|| * ||.v.||;
      end;
      suppose
        A12: ||.PL.|| <> 0;
        set a = ||.PL.||;
        A13: for y be VECTOR of V st y in Ker L
        holds (1/a) * ||.QL.v.|| <= ||.vv1+y.||
        proof
          let y be VECTOR of V;
          assume y in Ker L; then
          y in L"{0.W} by KLXY1,LCL1; then
          A14: y in the carrier of V & L.y in {0.W} by FUNCT_2:38;
          A15: QL.v = L.vv1 by A4,A9
                  .= L.vv1 + 0.W by RLVECT_1:4
                  .= L.vv1 + L.y by A14,TARSKI:def 1
                  .= L.(vv1+y) by VECTSP_1:def 20;
          (1/a) * ||.L.(vv1+y).|| <= (1/a) * (||.PL.|| * ||.vv1+y.||)
            by LOPBAN_1:32,XREAL_1:64; then
          (1/a) * ||.L.(vv1+y).|| <= (1/a) * ||.PL.|| * ||.vv1+y.||; then
          (1/a) * ||.L.(vv1+y).|| <= 1 * ||.vv1+y.|| by A12,XCMPLX_1:106;
          hence thesis by A15,SUBTH0;
        end;
        for r being ExtReal st r in NormVSets(V,Ker L,vv1)
        holds (1/a) * ||.QL.v.|| <= r
        proof
          let r be ExtReal;
          assume r in NormVSets(V,Ker L,vv1); then
          consider x be VECTOR of V such that
          A16: r = ||.x.|| & x in vv1 + Ker L;
          consider y be VECTOR of V such that
          A17: x = vv1+y & y in Ker L by A16;
          thus thesis by A13,A16,A17;
        end; then
        A18: (1/a) * ||.QL.v.|| is LowerBound of NormVSets(V,Ker L,vv1)
        by XXREAL_2:def 2;
        a * ((1/a) * ||.QL.v.||) <= a * ||.v.||
        by A10,A18,XREAL_1:64,XXREAL_2:def 4; then
        a * (1/a) * ||.QL.v.|| <= a * ||.v.||; then
        1 * ||.QL.v.|| <= a * ||.v.|| by A12,XCMPLX_1:106;
        hence ||.QL.v.|| <= ||.PL.|| * ||.v.||;
      end;
    end;
    reconsider QL as Lipschitzian LinearOperator of NVectQuot(V,Ker L), Im L
    by A8,LOPBAN_1:def 8;
    take QL;
    A19: R_NormSpace_of_BoundedLinearOperators(NVectQuot(V,Ker L), Im L)
      = NORMSTR (# BoundedLinearOperators(NVectQuot(V,Ker L), Im L),
                   Zero_(BoundedLinearOperators(NVectQuot(V,Ker L), Im L),
                   R_VectorSpace_of_LinearOperators(NVectQuot(V,Ker L), Im L)),
                   Add_(BoundedLinearOperators(NVectQuot(V,Ker L), Im L),
                   R_VectorSpace_of_LinearOperators(NVectQuot(V,Ker L), Im L)),
                   Mult_(BoundedLinearOperators(NVectQuot(V,Ker L), Im L),
                   R_VectorSpace_of_LinearOperators(NVectQuot(V,Ker L), Im L)),
                   BoundedLinearOperatorsNorm(NVectQuot(V,Ker L), Im L) #)
    by LOPBAN_1:def 14; then
    reconsider PQL = QL as Point of
    R_NormSpace_of_BoundedLinearOperators(NVectQuot(V,Ker L), Im L)
    by LOPBAN_1:def 9;
    A20: PreNorms(QL)
      = {||.QL.t.|| where t is VECTOR of NVectQuot(V,Ker L) : ||.t.|| <= 1}
      by LOPBAN_1:def 12;
    now
      let r be Real;
      assume r in PreNorms QL; then
      consider v be VECTOR of NVectQuot(V,Ker L) such that
      A21: r = ||.QL.v.|| and
      A22: ||.v.|| <= 1 by A20;
      A23: ||.QL.v.|| <= ||.PL.|| * ||.v.|| by A8;
      ||.PL.|| * ||.v.|| <= ||.PL.|| * 1 by A22,XREAL_1:64;
      hence r <= ||.PL.|| by A21,A23,XXREAL_0:2;
    end; then
    upper_bound PreNorms QL <= ||.PL.|| by SEQ_4:45; then
    A24: ||.PQL.|| <= ||.PL.|| by A19,LOPBAN_1:30;
    R_NormSpace_of_BoundedLinearOperators(V,W)
      = NORMSTR(# BoundedLinearOperators(V,W),
                  Zero_(BoundedLinearOperators(V,W),
                  R_VectorSpace_of_LinearOperators(V,W)),
                  Add_(BoundedLinearOperators(V,W),
                  R_VectorSpace_of_LinearOperators(V,W)),
                  Mult_(BoundedLinearOperators(V,W),
                  R_VectorSpace_of_LinearOperators(V,W)),
                  BoundedLinearOperatorsNorm(V,W) #) by LOPBAN_1:def 14; then
    A26: ||.PL.|| = upper_bound PreNorms(L) by LOPBAN_1:30;
    A27: PreNorms L is non empty bounded_above by LOPBAN_1:27;
    A28: PreNorms(L) = {||.L.t.|| where t is VECTOR of V : ||.t.|| <= 1}
        by LOPBAN_1:def 12;
    now
      let s be Real;
      assume 0 < s; then
      consider p be Real such that
      A29: p in PreNorms(L) & ||.PL.||- s < p by A26,A27,SEQ_4:def 1;
      consider vv1 be VECTOR of V such that
      A30: p = ||.L.vv1.|| and
      A31: ||.vv1.|| <= 1 by A28,A29;
      A32: lower_bound NormVSets(V,Ker L,vv1) <= ||.vv1.|| by LMQ23;
      reconsider v = vv1 + (Ker L) as Point of NVectQuot(V,Ker L) by A2,LMQ07;
      ||.v.|| = lower_bound NormVSets(V,Ker L,vv1) by A2,DeNorm; then
      A33: ||.v.|| <= 1 by A31,A32,XXREAL_0:2;
      QL.v = L.vv1 by A4,A5; then
      A34: ||.QL.v.|| = ||.L.vv1.|| by SUBTH0;
      A35: ||.QL.v.|| <= ||.PQL.|| * ||.v.|| by LOPBAN_1:32;
      ||.PQL.|| * ||.v.|| <= ||.PQL.|| * 1 by A33,XREAL_1:64; then
      ||.QL.v.|| <= ||.PQL.|| by A35,XXREAL_0:2; then
      ||.PL.|| - s <= ||.PQL.|| by A29,A30,A34,XXREAL_0:2; then
      ||.PL.||- s + s <= ||.PQL.|| + s by XREAL_1:6;
      hence ||.PL.|| <= ||.PQL.|| + 1 * s;
    end; then
    ||.PL.|| <= ||.PQL.|| by LMINEQ;
    hence thesis by A4,A5,A24,XXREAL_0:1;
  end;
