reserve X,Y,Z for non trivial RealBanachSpace;

theorem LM80:
  for I be PartFunc of
       R_NormSpace_of_BoundedLinearOperators(X,Y),
       R_NormSpace_of_BoundedLinearOperators(Y,X)
  st dom I = InvertibleOperators(X,Y)
   & for u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
      st u in InvertibleOperators(X,Y)
     holds I.u = Inv u
  holds I is_continuous_on InvertibleOperators(X,Y)
  proof
    let I be PartFunc of
         R_NormSpace_of_BoundedLinearOperators(X,Y),
         R_NormSpace_of_BoundedLinearOperators(Y,X);
    assume
    A1: dom I = InvertibleOperators(X,Y)
       & for u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
          st u in InvertibleOperators(X,Y)
         holds I.u = Inv u;
    set S = R_NormSpace_of_BoundedLinearOperators(X,Y);
    set T = InvertibleOperators(X,Y);
    for x0 being Point of S
    for r being Real st x0 in T & 0 < r holds
    ex s being Real
    st 0 < s
     & for x1 being Point of S st x1 in T & ||.x1 - x0.|| < s
       holds ||.(I /. x1) - (I /. x0).|| < r
    proof
      let x0 be Point of S;
      let r be Real;
      assume
      A2: x0 in T & 0 < r; then
      ex u be Point of S st x0 = u & u is invertible;
      then consider s be Real such that
      A4: 0 < s
        & for v be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
          st ||.v-x0.|| < s
          holds ||.Inv v - Inv x0 .|| < r by A2,LMTh3;
      take s;
      thus 0 < s by A4;
      let x1 be Point of S;
      assume
      A5: x1 in T & ||.x1 - x0.|| < s;
      A7: I/.x0 = I.x0 by A1,A2,PARTFUN1:def 6
               .= Inv x0 by A1,A2;
      I/.x1 = I.x1 by A1,A5,PARTFUN1:def 6
           .= Inv x1 by A1,A5;
      hence thesis by A4,A5,A7;
    end;
    hence I is_continuous_on InvertibleOperators(X,Y) by A1,NFCONT_1:19;
  end;
