 reserve S,T,W,Y for RealNormSpace;
 reserve f,f1,f2 for PartFunc of S,T;
 reserve Z for Subset of S;
 reserve i,n for Nat;
 reserve X,Y,Z for non trivial RealBanachSpace;

theorem LMTh3:
  for u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
  st u is invertible
  holds
    ex K,s be Real
    st 0 <= K & 0 < s
     & for du be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
       st ||.du.|| < s
       holds
         u+du is invertible
       & ||.Inv(u+du) - Inv u - (- (Inv u)*du*(Inv u) ) .||
          <= K * (||.du.|| * ||.du.||)
  proof
    let u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
    assume
    A1: u is invertible;
    set s1 = 1 / ||.Inv u .||;
    set AG = R_Normed_Algebra_of_BoundedLinearOperators X;
    A3: 0 < ||. Inv u .|| by A1,LOPBAN13:12;
    set s2 = (1/2) / ||. Inv u .||;
    A5: 0 < s2 by A3,XREAL_1:139;
    set s12 = min(s1,s2);
    A7: 0 < s12 & s12 <= s1 & s12 <= s2 by A3,A5,XXREAL_0:15,17;
    set K = 2 * ||. (Inv u) .|| * ||. (Inv u) .|| * ||. (Inv u) .||;
    take K,s12;
    thus 0 <= K & 0 < s12 by A3,A5,XXREAL_0:15;
    let du be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
    assume
    A8: ||.du.|| < s12; then
    A9: ||.du.|| < s1 by A7,XXREAL_0:2;
    hence u+du is invertible by A1,LOPBAN13:13;
    consider w be Point of R_Normed_Algebra_of_BoundedLinearOperators X,
           s,I be Point of R_NormSpace_of_BoundedLinearOperators(X,X)
    such that
    A11: w = (Inv u) * du
      & 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 + du) = Inv(I+s) * (Inv u) by A1,A9,LOPBAN13:13;
    reconsider sA = s as Point of AG;
    A13: I * (Inv u)
     = (id the carrier of X) * modetrans(Inv u,Y,X) by A11,LOPBAN_1:def 11
    .= modetrans(Inv u,Y,X) by FUNCT_2:17
    .= Inv u by LOPBAN_1:def 11;
    reconsider IIu = I * (Inv u)
      as Point of R_NormSpace_of_BoundedLinearOperators(Y,X);
    set L = ( Inv u ) * du * ( Inv u );
    A14: Inv (u+du) - Inv u -( -( Inv u) * du * (Inv u) )
      = Inv(I+s) * (Inv u) - IIu + L by A11,A13,RLVECT_1:17
     .= ( Inv(I+s) -I) * (Inv u) + L by LOPBAN13:20;
    A15: Inv(I+s)*(I+s) = I by A11,LOPBAN13:22;
    Inv(I+s)*I = modetrans(Inv(I+s),X,X) * (id the carrier of X)
        by A11,LOPBAN_1:def 11
    .= modetrans(Inv(I+s),X,X) by FUNCT_2:17
    .= Inv(I+s) by LOPBAN_1:def 11; then
    A17: Inv(I+s) - I = Inv(I+s) * (I - (I+s) ) by A15,LOPBAN13:19
    .= Inv(I+s) * (- ((Inv u) * du)) by A11,LOPBAN13:21; then
    A19: (Inv(I+s) -I) * (Inv u)
     = ( - Inv(I+s) * ((Inv u) * du)) * Inv u by LOPBAN13:26
    .= - ( Inv(I+s) * ((Inv u) * du) * Inv u) by LOPBAN13:26
    .= - Inv(I+s) * L by LOPBAN13:10;
    I * L = (id the carrier of X) * modetrans(L,Y,X) by A11,LOPBAN_1:def 11
    .= modetrans(L,Y,X) by FUNCT_2:17
    .= L by LOPBAN_1:def 11;
    then - Inv(I+s) * L + L = I * L - Inv(I+s) * L
    .= (I - Inv(I+s)) * L by LOPBAN13:20
    .= (- (Inv(I+s)-I) ) * L by RLVECT_1:33
    .= - Inv(I+s) * ( - ((Inv u) * du) ) * ((Inv u) * du * (Inv u))
       by A17,LOPBAN13:26; then
    Inv (u+du) - Inv u -( -(Inv u) * du * (Inv u) )
     = - ((- ( Inv(I+s) *((Inv u) * du)))*((Inv u)*du*(Inv u)))
        by A14,A19,LOPBAN13:26
    .= - - (Inv(I+s)*((Inv u) * du)) * ((Inv u)*du*(Inv u)) by LOPBAN13:26
    .= (Inv(I+s) * ((Inv u) * du))*((Inv u)*du*(Inv u)) by RLVECT_1:17;
    then
    A22: ||. Inv(u+du) - Inv u -(-(Inv u)*du*(Inv u)) .||
     <= ||. Inv(I+s) * ((Inv u) * du) .|| * ||. (Inv u)*du*(Inv u) .||
      by LOPBAN13:18;
    A23: ||. Inv(I+s) * ((Inv u) * du) .||
      <= ||. Inv(I+s) .|| * ||. (Inv u) * du .|| by LOPBAN13:18;
    ||. Inv(I+s) .|| * ||. (Inv u) * du .||
      <= ||. Inv(I+s) .|| * (||. (Inv u).|| * ||.du .||)
        by LOPBAN13:18,XREAL_1:64; then
    A25: ||. Inv(I+s) * ((Inv u) * du) .||
      <= ||. Inv(I+s) .|| * ( ||. (Inv u).|| * ||.du .|| ) by A23,XXREAL_0:2;
    A26: ||. (Inv u) * du .|| * ||.Inv u .||
      <= ||. (Inv u).|| * ||.du .|| * ||.Inv u .|| by LOPBAN13:18,XREAL_1:64;
    ||. (Inv u)*du*(Inv u) .||
      <= ||. (Inv u)*du .|| * ||.Inv u .|| by LOPBAN13:18; then
    ||. (Inv u)*du*(Inv u) .||
      <= ||. (Inv u).|| * ||.du .|| * ||.Inv u .|| by A26,XXREAL_0:2; then
    ||. Inv(I+s) *( (Inv u) * du ) .|| * ||. ( Inv u )*du*(Inv u ) .||
      <= ||. Inv(I+s) .|| * ( ||. (Inv u).|| * ||.du .|| )
        * (||. (Inv u).|| * ||.du .|| * ||.Inv u .|| )
        by A25,XREAL_1:66; then
    A28: ||. Inv(u+du) - Inv u -( -(Inv u)*du*(Inv u) ) .||
      <= ||. Inv(I+s) .|| * ( ||. (Inv u).|| * ||.du .|| )
        * (||. (Inv u).|| * ||.du .|| * ||.Inv u .||) by A22,XXREAL_0:2;
    A29: ||.s.|| <= ||. Inv u .|| * ||.du.|| by A11,LOPBAN13:18;
    ||.du.|| < s2 by A7,A8,XXREAL_0:2; then
    ||. Inv u .|| * ||.du.||
      <= ||. Inv u .|| * ((1/2) / ||. Inv u .||) by XREAL_1:64; then
    ||. Inv u .|| * ||.du.|| <= 1/2 by A3,XCMPLX_1:87; then
    ||.s.|| <= 1/2 by A29,XXREAL_0:2; then
    1-1/2 <= 1-||.s.|| by XREAL_1:10; then
    1 /(1-||.s.||) <= 1/(1/2) by XREAL_1:118; then
    ||.Inv (I+s).|| <=2 by A11,XXREAL_0:2; then
    ||. Inv(I+s) .|| * ((||. (Inv u).|| * ||.du .||)
      * (||. (Inv u).|| * ||.du .|| * ||.Inv u .||))
      <= 2 * ((||. (Inv u).|| * ||.du .||)
        * (||. (Inv u).|| * ||.du .|| * ||.Inv u .||)) by XREAL_1:64;
    hence ||. Inv (u+du) - Inv u -( -(Inv u)*du*(Inv u) ) .||
       <= K * (||.du .|| * ||.du .||) by A28,XXREAL_0:2;
  end;
