
theorem LMQ21:
  for V,W be RealLinearSpace, L be LinearOperator of V,W holds
  ex QL be LinearOperator of VectQuot(V,Ker L), Im L
  st QL is isomorphism
   & for z be Point of VectQuot(V,Ker L), v be VECTOR of V
     st z = v + Ker L holds QL.z = L.v
  proof
    let V,W be RealLinearSpace, L be LinearOperator of V,W;
    A1: the carrier of Im(L) = rng L by IMX2,LCL1;
    A2: the carrier of Ker(L) = L"{0.W} by KLXY1,LCL1;
    defpred P[object,object] means
    ex v be VECTOR of V st $1 = v + Ker L & $2 = L.v;
    A3: for x being Element of the carrier of VectQuot(V,Ker L)
    ex y being Element of the carrier of (Im L) st P[x,y]
    proof
      let x be Element of the carrier of VectQuot(V,Ker L);
      consider v be Point of V such that
      A4: x = v + Ker L by LMQ07;
      reconsider y = L.v as Element of the carrier of Im L by A1,FUNCT_2:4;
      take y;
      thus thesis by A4;
    end;
    consider QL being Function of the carrier of VectQuot(V,Ker L), the
    carrier of Im L such that
    A5: for x being Element of VectQuot(V,Ker L) holds P[x,QL.x]
    from FUNCT_2:sch 3(A3);
    A6: for z be Point of VectQuot(V,Ker L), v be VECTOR of V st z = v + Ker L
    holds QL.z = L.v
    proof
      let z be Point of VectQuot(V,Ker L), v be VECTOR of V;
      assume
      A7: z = v + Ker L;
      consider w be VECTOR of V such that
      A8: z = w + Ker L & QL.z = L.w by A5;
      consider v1 be Point of V such that
      A9: v1 in Ker L & w = v + v1 by A7,A8,RLSUB_1:54,63;
      w - v = v1 + (v -v) by A9,RLVECT_1:28
           .= v1 + 0.V by RLVECT_1:15
           .= v1 by RLVECT_1:4; then
      A10: w - v in the carrier of V & L.(w-v) in {0.W} by A2,A9,FUNCT_2:38;
      L.(w-v) = L.(w+(-1)*v) by RLVECT_1:16
             .= L.w + L.((-1)*v) by VECTSP_1:def 20
             .= L.w + (-1)* L.v by LOPBAN_1:def 5
             .= L.w + - L.v by RLVECT_1:16; then
      L.w - L.v = 0.W by A10,TARSKI:def 1; then
      L.v = L.w - L.v + L.v by RLVECT_1:4
         .= L.w - (L.v - L.v) by RLVECT_1:29
         .= L.w - 0.W by RLVECT_1:15
         .= L.w by RLVECT_1:13;
      hence QL.z = L.v by A8;
    end;
    A11: for x1,x2 being object
    st x1 in the carrier of VectQuot(V,Ker L)
     & x2 in the carrier of VectQuot(V,Ker L)
     & QL.x1 = QL.x2
    holds x1 = x2
    proof
      let x1,x2 be object;
      assume
      A12: x1 in the carrier of VectQuot(V,Ker L)
        & x2 in the carrier of VectQuot(V,Ker L)
        & QL.x1 = QL.x2;
      reconsider z1 = x1, z2 = x2 as Point of VectQuot(V,Ker L) by A12;
      consider v1 be VECTOR of V such that
      A13: z1 = v1 + Ker L & QL.x1 = L.v1 by A5;
      consider v2 be VECTOR of V such that
      A14: z2 = v2 + Ker L & QL.x2 = L.v2 by A5;
      L.v1 - L.v2 = L.v1 + (-1) * L.v2 by RLVECT_1:16
                 .= L.v1 + L.((-1) * v2) by LOPBAN_1:def 5
                 .= L.(v1 + (-1) * v2) by VECTSP_1:def 20
                 .= L.(v1 - v2) by RLVECT_1:16; then
      L.(v1-v2) = 0.W by A12,A13,A14,RLVECT_1:15; then
      L.(v1-v2) in {0.W} by TARSKI:def 1; then
      A15: v1-v2 in Ker(L) by A2,FUNCT_2:38;
      (v1-v2) + v2 = v1 - (v2-v2) by RLVECT_1:29
                  .= v1 - 0.V by RLVECT_1:15
                  .= v1 by RLVECT_1:13; then
      v1 in v2 + Ker(L) by A15;
      hence thesis by A13,A14,RLSUB_1:54;
    end;
    for v being object st v in the carrier of Im(L)
    ex s being object st s in the carrier of VectQuot(V,Ker L) & v = QL.s
    proof
      let v be object;
      assume v in the carrier of Im(L); then
      v in rng L by IMX2,LCL1; then
      consider x be object such that
      A16: x in the carrier of V & L.x = v by FUNCT_2:11;
      reconsider x as Point of V by A16;
      reconsider s = x + Ker L as Point of VectQuot(V,Ker L) by LMQ07;
      take s;
      thus s in the carrier of VectQuot(V,Ker L) & v = QL.s by A6,A16;
    end; then
    A17: QL is onto by FUNCT_2:10;
    A18: for v,w be Element of VectQuot(V,Ker L) holds QL.(v+w) = QL.v + QL.w
    proof
      let v,w be Element of VectQuot(V,Ker L);
      consider x be Point of V such that
      A19: v = x + Ker L by LMQ07;
      consider y be Point of V such that
      A20: w = y + Ker L by LMQ07;
      A21: v + w = (x + y) + Ker L by A19,A20,LMQ11;
      A22: QL.v = L.x by A6,A19;
      A23: QL.w = L.y by A6,A20;
      thus QL.(v + w) = L.(x + y) by A6,A21
                     .= L.x + L.y by VECTSP_1:def 20
                     .= QL.v + QL.w by A22,A23,RLSUB_1:13;
    end;
    for v being VECTOR of VectQuot(V,Ker L), r being Real
    holds QL.(r*v) = r * QL.v
    proof
      let v be VECTOR of VectQuot(V,Ker L), r being Real;
      consider x be Point of V such that
      A24: v = x + Ker L by LMQ07;
      r*v = (r*x) + Ker L by A24,LMQ09;
      hence QL.(r*v) = L.(r*x) by A6
                    .= r * L.x by LOPBAN_1:def 5
                    .= r * QL.v by A6,A24,RLSUB_1:14;
    end; then
    QL is additive homogeneous by A18,LOPBAN_1:def 5; then
    reconsider QL as LinearOperator of VectQuot(V,Ker L), Im L;
    take QL;
    thus QL is isomorphism by A11,A17,FUNCT_2:19;
    thus thesis by A6;
  end;
