reserve X,Y,Z for non trivial RealBanachSpace;

theorem Th2:
  for u,v be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
  st u is invertible
   & ||.v.|| < 1 / ||. Inv u .||
  holds u+v is invertible
     & ||.Inv (u+v).|| <= 1 / ( 1 / ||.Inv (u).|| - ||.v.|| )
     & ex w be Point of R_Normed_Algebra_of_BoundedLinearOperators X,
        s,I be Point of R_NormSpace_of_BoundedLinearOperators(X,X)
       st w = (Inv u) * v
        & s = w
        & I = id X
        & ||.s.|| < 1
        & (-w) GeoSeq is norm_summable
        & I+s is invertible
        & ||.Inv(I+s).|| <= 1 / ( 1 - ||.s.|| )
        & Inv(I+s) = Sum ( (-w) GeoSeq )
        & Inv(u+v) = Inv(I+s) * (Inv u)
  proof
    let u,v be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
    assume that
    A1: u is invertible and
    A2: ||.v.|| < 1 / ||.Inv u .||;
    set S = R_Normed_Algebra_of_BoundedLinearOperators X;
    1.S = id X; then
    reconsider I = id X
      as Point of R_NormSpace_of_BoundedLinearOperators(X,X);
    reconsider Is = I as Point of S;
    A6: u is Lipschitzian LinearOperator of X,Y by LOPBAN_1:def 9;
    A7: v is Lipschitzian LinearOperator of X,Y by LOPBAN_1:def 9;
    reconsider udv = (Inv u) * v
      as Point of R_NormSpace_of_BoundedLinearOperators(X,X);
    reconsider udv2 = udv as Point of S;
    reconsider Lu = u,Lv = v
      as Lipschitzian LinearOperator of X,Y by LOPBAN_1:def 9;
    reconsider LInvu = Inv u
      as Lipschitzian LinearOperator of Y,X by LOPBAN_1:def 9;
    A14: modetrans (LInvu,Y,X) = LInvu & modetrans (Lv,X,Y) = Lv
      by LOPBAN_1:29; then
    A15: ||.udv.|| <= ||. Inv u .|| * ||.v.|| by LOPBAN_2:2;
    LInvu = u" by A1,Def1; then
    A17: BoundedLinearOperatorsNorm(X,X).(LInvu*Lu)
     = ||.1.S.|| by A1,FUNCT_2:29
    .= 1 by LOPBAN_2:def 10;
    A18: ||.Inv u.|| <> 0
    proof
      assume ||.Inv u.|| = 0; then
      1 <= 0 * (BoundedLinearOperatorsNorm(X,Y).u) by A17,LOPBAN_2:2;
      hence contradiction;
    end; then
    ||.Inv u .|| * ||.v.|| < ||. Inv u .|| * ( 1 / ||. Inv u .|| )
      by A2,XREAL_1:68; then
    A20: ||.Inv u .|| * ||.v.|| < 1 by A18,XCMPLX_1:106; then
    A21: ||.udv.|| < 1 by A15,XXREAL_0:2; then
    A22: I + udv is invertible
       & ||.Inv (I+udv).|| <= 1 / ( 1 - ||.udv.|| )
       & ex w be Point of R_Normed_Algebra_of_BoundedLinearOperators X
         st w = udv
          & (-w) GeoSeq is norm_summable
          & Inv (I+udv) = Sum ( (-w) GeoSeq ) by Th1;
    A23: u+v is Lipschitzian LinearOperator of X,Y by LOPBAN_1:def 9; then
    A24: modetrans(u+v,X,Y) = u+v by LOPBAN_1:29;
    A25: I+udv is Lipschitzian LinearOperator of X,X
         by LOPBAN_1:def 9; then
    A26: modetrans(I+udv,X,X) = I+udv by LOPBAN_1:29;
    A27: modetrans(u,X,Y) = u by A6,LOPBAN_1:29;
    A28: for x be Element of the carrier of X
         holds (u+v).x = (u*(I+udv)).x
    proof
      let x be Element of the carrier of X;
      A33: (u*(I+udv)).x
        = modetrans(u,X,Y).(modetrans(I+udv,X,X).x) by FUNCT_2:15
       .= u.((I+udv).x) by A25,A27,LOPBAN_1:29;
      A35: (I+udv).x = (id X).x + udv.x by LOPBAN_1:35
       .= x + udv.x;
      Lu is additive; then
      A37: Lu.((I+udv).x) = Lu.x + Lu.(udv.x) by A35;
      A39: Inv u is Lipschitzian LinearOperator of Y,X by LOPBAN_1:def 9;
      A40: modetrans(v,X,Y) = v by A7,LOPBAN_1:29;
      u.(udv.x) = u.(modetrans(Inv u,Y,X).(modetrans(v,X,Y).x)) by FUNCT_2:15
      .= u.((Inv u).(v.x)) by A39,A40,LOPBAN_1:29
      .= u.(u".(v.x)) by A1,Def1
      .= v.x by A1,FUNCT_1:35;
      hence thesis by A33,A37,LOPBAN_1:35;
    end;
    then A43: u+v is one-to-one by A1,A22,A23,A26,A27,FUNCT_2:def 7;
    A44: modetrans(u,X,Y) is onto by A1,A6,LOPBAN_1:29;
    modetrans(I+udv,X,X) is onto by A22,A25,LOPBAN_1:29; then
    modetrans(u,X,Y) * modetrans(I+udv,X,X) is onto by A44,FUNCT_2:27; then
    A46: rng (u+v) = the carrier of Y by A23,A28,FUNCT_2:def 7;
    set Iuv = Inv (I+udv) * (Inv u);
    Iuv is Lipschitzian LinearOperator of Y,X by LOPBAN_2:2; then
    A48: modetrans(Iuv,Y,X) = Iuv by LOPBAN_1:29;
    Inv u is Lipschitzian LinearOperator of Y,X by LOPBAN_1:def 9; then
    A49: modetrans(Inv u,Y,X) = Inv u by LOPBAN_1:29;
    B49: Inv (I+udv) is Lipschitzian LinearOperator of X,X
      by LOPBAN_1:def 9; then
    A50: modetrans(Inv(I+udv),X,X) = Inv(I+udv) by LOPBAN_1:29;
    A51: modetrans((I+udv)",X,X )
       = modetrans(Inv (I+udv),X,X) by A21,Def1,Th1
      .= Inv (I+udv) by B49,LOPBAN_1:29
      .= (I+udv)" by A21,Def1,Th1;
    modetrans(Inv u,Y,X) = u" by A1,A49,Def1; then
    A53: (Inv u) * u = u" * u by A6,LOPBAN_1:29
      .= id X by A1,FUNCT229;
    (Inv u) * u is Lipschitzian LinearOperator of X,X by LOPBAN_2:2; then
    modetrans(((Inv u) * u),X,X) = id X by A53,LOPBAN_1:29;
    then A55: ((Inv u) * u)*(I+udv)
     = (id X) * (I+udv) by A25,LOPBAN_1:29
    .= (I+udv) by A25,FUNCT_2:17;
    A56: modetrans(Iuv,Y,X)*modetrans(u+v,X,Y)
       = (Inv(I+udv) * (Inv u)) *(u+v)
      .= Inv(I+udv) * ((Inv u) * (u+v)) by RELAT136
      .= Inv(I+udv) * ((Inv u) * (u*(I+udv))) by A23,A28,FUNCT_2:def 7
      .= Inv(I+udv) * (I+udv) by A55,RELAT136
      .= modetrans((I+udv)",X,X ) * modetrans(I+udv,X,X) by A21,Def1,Th1
      .= (I+udv)" * (I+udv) by A25,A51,LOPBAN_1:29
      .= id X by A22,FUNCT229;
    then A57: modetrans(u+v,X,Y) " = modetrans(Iuv,Y,X)
      by A24,A43,A46,FUNCT_2:30;
    thus
    A58: u+v is invertible by A24,A43,A46,A48,A56,FUNCT_2:30;
    reconsider Iuvp = Iuv
      as Point of R_NormSpace_of_BoundedLinearOperators(Y,X);
    A59: BoundedLinearOperatorsNorm(Y,X).(Iuv)
     <= (BoundedLinearOperatorsNorm(X,X).modetrans(Inv (I+udv),X,X))
       *(BoundedLinearOperatorsNorm(Y,X).modetrans((Inv u),Y,X))
      by LOPBAN_2:2;
    (BoundedLinearOperatorsNorm(X,X). Inv(I+udv))
        * (BoundedLinearOperatorsNorm(Y,X).(Inv u))
      <= (1 / (1 - ||.udv.||)) * ||.(Inv u).||
        by A22,XREAL_1:64; then
    A64: ||.Iuvp.|| <= (1 / (1 - ||.udv.||)) * ||.(Inv u).||
      by A49,A50,A59,XXREAL_0:2;
    A65: 1 - (||.(Inv u).|| * ||.v.||) <= 1 - ||.udv.||
      by A14,LOPBAN_2:2,XREAL_1:10;
    0 < 1 - (||.(Inv u).|| * ||.v.||) by A20,XREAL_1:50; then
    1 / (1 - ||.udv.||) <= 1 / (1- ||.(Inv u).|| * ||.v.||)
          by A65,XREAL_1:118; then
    1 / (1 - ||.udv.||) * ||.(Inv u).||
      <= 1 / (1 - ||.(Inv u).|| * ||.v.||) * ||.(Inv u).||
        by XREAL_1:64; then
    A67: 1 / (1 - ||.udv.||) * ||.(Inv u).||
      <= ||.(Inv u).|| / (1 - ||.(Inv u).|| * ||.v.||) by XCMPLX_1:99;
    (1 - ||.(Inv u).|| * ||.v.||) / ||.(Inv u).||
     = 1 / ||.(Inv u).|| - (||.(Inv u).|| * ||.v.||) / ||.(Inv u).||
        by XCMPLX_1:120
    .= 1 / ||.(Inv u).|| - ||.v.|| by A18,XCMPLX_1:89; then
    1 / (1 / ||.(Inv u).|| - ||.v.||)
        = ||.(Inv u).|| / (1 - ||.(Inv u).|| * ||.v.||) by XCMPLX_1:57; then
    ||.Iuvp.|| <= 1 / (1 / ||.(Inv u).|| - ||.v.||)
      by A64,A67,XXREAL_0:2;
    hence ||.Inv (u+v).|| <= 1 / (1 / ||.Inv (u).|| - ||.v.||)
      by A24,A48,A57,A58,Def1;
    consider w be Point of R_Normed_Algebra_of_BoundedLinearOperators X
    such that
    A70: w = udv
     & (-w) GeoSeq is norm_summable
     & Inv(I+udv) = Sum((-w) GeoSeq) by A21,Th1;
    reconsider S = Sum((-w) GeoSeq)
      as Point of R_NormSpace_of_BoundedLinearOperators(X,X);
    take w,udv,I;
    thus w = (Inv u) * v
      & udv = w
      & I = id X
      & ||.udv.|| < 1
      & (-w) GeoSeq is norm_summable
      & I+udv is invertible
      & ||.Inv (I+udv).|| <= 1 / ( 1 - ||.udv.|| )
      & Inv (I+udv) = Sum ( (-w) GeoSeq )
        by A21,A70,Th1;
    thus Inv (u+v) = Inv (I+udv) * (Inv u) by A24,A48,A57,A58,Def1;
  end;
