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
    for r be Real st 0 < r
    ex s be Real
    st 0 < s
     & for v be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
       st ||.v-u.|| < s
       holds ||.Inv v - Inv u .|| < r
  proof
    let u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
    assume
    A1: u is invertible;
    let r be Real;
    assume
    A2: 0 < r;
    set r0 = r/2;
    A3: 0 < r0 & r0 < r by A2,XREAL_1:215,216;
    set s1 = 1 / ||.Inv u .||;
    set AG = R_Normed_Algebra_of_BoundedLinearOperators X;
    A5: 0 < ||. Inv u .|| by A1,LM50; then
    A6: 0 < s1 by XREAL_1:139;
    set s2 = (1/2) / ||. Inv u .||;
    A7: 0 < s2 by A5,XREAL_1:139;
    A8: 0 < ||. Inv u .|| * ||. Inv u .|| by A5,XREAL_1:129;
    A9: 0 < r0/2 by A3,XREAL_1:215;
    set s3 = (r0/2) / (||. Inv u .|| * ||. Inv u .||);
    A10: 0 < s3 by A8,A9,XREAL_1:139;
    set s4 = min(s1,s2);
    A11: 0 < s4 & s4 <= s1 & s4 <= s2 by A6,A7,XXREAL_0:15,17;
    set s = min(s4,s3);
    B11: 0 < s & s <= s4 & s <= s3 by A10,A11,XXREAL_0:15,17; then
    A12: 0 < s & s <= s1 & s <= s2 & s <= s3 by A11,XXREAL_0:2;
    take s;
    thus 0 < s by A10,A11,XXREAL_0:15;
    let v be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
    assume
    A13: ||.v-u.|| < s; then
    ||.v-u.|| < s1 by A12,XXREAL_0:2; then
    consider w be Point of R_Normed_Algebra_of_BoundedLinearOperators X,
           s,I be Point of R_NormSpace_of_BoundedLinearOperators(X,X)
    such that
    A14: w = (Inv u) * (v-u)
      & 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 v = Inv(I+s) * (Inv u) by A1,LMTh2;
    reconsider sA = s as Point of AG;
    A16: I * (Inv u) = (id X)*modetrans(Inv u,Y,X) by A14,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);
    Inv v - Inv u = (Inv(I+s) - I) * (Inv u) by A14,A16,LM200; then
    A18: ||. Inv v - Inv u.|| <= ||. Inv(I+s) -I .|| * ||. Inv u .|| by NRM;
    A19: Inv(I+s)*(I+s) = I by A14,LM400;
    Inv(I+s)*I = modetrans(Inv(I+s),X,X) * (id X)
        by A14,LOPBAN_1:def 11
    .= modetrans(Inv(I+s),X,X) by FUNCT_2:17
    .= Inv(I+s) by LOPBAN_1:def 11;
    then Inv(I+s) - I = Inv(I+s) * (I - (I+s)) by A19,LM100
    .= Inv(I+s) * (-s) by LM300; then
    ||. Inv(I+s) -I .|| <= ||.Inv(I+s) .|| * ||.-s.|| by NRM; then
    A23: ||. Inv(I+s) -I .|| <= ||.Inv(I+s) .|| * ||.s.|| by NORMSP_1:2;
    A24: ||.s.|| <= ||. Inv u .|| * ||.v-u.|| by A14,NRM;
    ||.Inv(I+s) .|| * ||.s.||
      <= ||.Inv(I+s) .|| * (||. Inv u .|| * ||.v-u.|| )
      by A14,NRM,XREAL_1:64; then
    A25: ||. Inv(I+s) -I .||
      <= ||.Inv(I+s) .|| * ||. Inv u .|| * ||.v-u.|| by A23,XXREAL_0:2;
    ||.Inv(I+s) .|| * ( ||. Inv u .|| * ||.v-u.|| )
      <= ( 1 / ( 1 - ||.s.|| )) * ( ||. Inv u .|| * ||.v-u.|| )
      by A14,XREAL_1:64; then
    ||. Inv(I+s) -I .||
      <= ( 1 / ( 1 - ||.s.|| )) * ||. Inv u .|| * ||.v-u.||
      by A25,XXREAL_0:2; then
    ||. Inv(I+s) -I .|| * ||. Inv u .||
      <= (( 1 / ( 1 - ||.s.|| )) * ||. Inv u .|| * ||.v-u.||)
        * ||. Inv u .|| by XREAL_1:64; then
    A29: ||. Inv v - Inv u.||
       <= (( 1 / ( 1 - ||.s.|| )) * ||. Inv u .|| * ||.v-u.||)
          * ||. Inv u .|| by A18,XXREAL_0:2;
    ||.v-u.|| < s2 by A12,A13,XXREAL_0:2; then
    ||. Inv u .|| * ||.v-u.||
       <= ||. Inv u .|| * ( (1/2) / ||. Inv u .|| ) by XREAL_1:64; then
    ||. Inv u .|| * ||.v-u.|| <= 1/2 by A5,XCMPLX_1:87; then
    ||.s.|| <= 1/2 by A24,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
A32: ( 1/ ( 1 - ||.s.|| )) * (||. Inv u .|| * ||. Inv u .|| * ||.v-u.|| )
      <= 2 * ( ||. Inv u .|| * ||. Inv u .|| * ||.v-u.|| ) by XREAL_1:64;
    ||.v-u.|| < s3 by B11,A13,XXREAL_0:2; then
    ||.v-u.|| * ( ||. Inv u .|| * ||. Inv u .|| )
      <= s3 * ( ||. Inv u .|| * ||. Inv u .|| ) by XREAL_1:64; then
    ||.v-u.|| * ( ||. Inv u .|| * ||. Inv u .|| )
      <= r0/2 by A8,XCMPLX_1:87; then
    2 * ( ||. Inv u .|| * ( ||. Inv u .|| * ||.v-u.|| ) )
      <= r0/2 * 2 by XREAL_1:64; then
    ( 1 / ( 1 - ||.s.|| )) * ( ||. Inv u .|| * ||. Inv u .|| * ||.v-u.|| )
      <= r0 by A32,XXREAL_0:2; then
    ||. Inv v - Inv u.|| <= r0 by A29,XXREAL_0:2;
    hence ||. Inv v - Inv u.|| < r by A3,XXREAL_0:2;
  end;
