reserve X,Y,Z for non trivial RealBanachSpace;

theorem
  ex I be PartFunc of
         R_NormSpace_of_BoundedLinearOperators(X,Y),
         R_NormSpace_of_BoundedLinearOperators(Y,X)
  st dom I = InvertibleOperators(X,Y)
   & rng I = InvertibleOperators(Y,X)
   & I is one-to-one
   & I is_continuous_on InvertibleOperators(X,Y)
   & ( ex J be PartFunc of
               R_NormSpace_of_BoundedLinearOperators(Y,X),
               R_NormSpace_of_BoundedLinearOperators(X,Y)
       st J = I"
        & J is one-to-one
        & dom J = InvertibleOperators(Y,X)
        & rng J = InvertibleOperators(X,Y)
        & J is_continuous_on InvertibleOperators(Y,X))
    & for u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
      st u in InvertibleOperators(X,Y)
      holds I.u = Inv u
  proof
    set S = R_NormSpace_of_BoundedLinearOperators(X,Y);
    set K = R_NormSpace_of_BoundedLinearOperators(Y,X);
    consider J be Function of InvertibleOperators(X,Y),InvertibleOperators(Y,X)
    such that
    A1: J is one-to-one
      & J is onto
      & for u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
         st u in InvertibleOperators(X,Y)
        holds J.u = Inv u by LM70;
    A2: InvertibleOperators(X,Y) <> {} implies
        InvertibleOperators(Y,X) <> {}
    proof
      assume InvertibleOperators(X,Y) <> {}; then
      consider x be object such that
      A3: x in InvertibleOperators(X,Y) by XBOOLE_0:def 1;
      consider u be Point of S such that
      A4: x=u & u is invertible by A3;
      Inv u is invertible by A4,LM60; then
      Inv u in InvertibleOperators(Y,X);
      hence InvertibleOperators(Y,X) <> {};
    end;
    then dom J = InvertibleOperators(X,Y) by FUNCT_2:def 1; then
    J in PFuncs (the carrier of R_NormSpace_of_BoundedLinearOperators(X,Y),
                 the carrier of R_NormSpace_of_BoundedLinearOperators(Y,X))
      by A1,PARTFUN1:def 3; then
    reconsider I = J
      as PartFunc of the carrier of R_NormSpace_of_BoundedLinearOperators(X,Y),
                     the carrier of R_NormSpace_of_BoundedLinearOperators(Y,X)
      by PARTFUN1:46;
    take I;
    thus
    A9: dom I = InvertibleOperators(X,Y) by A2,FUNCT_2:def 1;
    thus rng I = InvertibleOperators(Y,X) by A1;
    thus I is one-to-one by A1;
    reconsider L = J"
      as Function of InvertibleOperators(Y,X), InvertibleOperators(X,Y)
      by A1,FUNCT_2:25;
    A14: dom (J") = InvertibleOperators(Y,X) by A1,FUNCT_1:33;
    A16: rng (J") = dom J by A1,FUNCT_1:33
     .= InvertibleOperators(X,Y) by A2,FUNCT_2:def 1; then
    L in PFuncs (the carrier of R_NormSpace_of_BoundedLinearOperators(Y,X),
                 the carrier of R_NormSpace_of_BoundedLinearOperators(X,Y))
           by A14,PARTFUN1:def 3; then
    reconsider L
      as PartFunc of the carrier of R_NormSpace_of_BoundedLinearOperators(Y,X),
                     the carrier of R_NormSpace_of_BoundedLinearOperators(X,Y)
      by PARTFUN1:46;
    for v be Point of R_NormSpace_of_BoundedLinearOperators(Y,X)
     st v in InvertibleOperators(Y,X)
    holds L.v = Inv v
    proof
      let v be Point of R_NormSpace_of_BoundedLinearOperators(Y,X);
      assume v in InvertibleOperators(Y,X); then
      consider u be object such that
      A23: u in InvertibleOperators(X,Y) & J.u = v by A1,FUNCT_2:11;
      reconsider u
        as Point of R_NormSpace_of_BoundedLinearOperators(X,Y) by A23;
      A24: ex u0 be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
           st u = u0 & u0 is invertible by A23;
      Inv v = Inv (Inv u) by A1,A23
           .= u by A24,LM60;
      hence thesis by A1,A2,A23,FUNCT_2:26;
    end;
    hence thesis by A1,A9,A14,A16,LM80;
  end;
