reserve X,Y,Z for non trivial RealBanachSpace;

theorem LM70:
  ex I be Function of InvertibleOperators(X,Y),InvertibleOperators(Y,X)
  st I is one-to-one
   & I is onto
   & 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);
    defpred P1[object,object] means
    ex u be Point of S
    st $1 = u & $2 = Inv u;
    A1: for x being object st x in InvertibleOperators(X,Y) holds
        ex y being object st y in InvertibleOperators(Y,X) & P1[x,y]
    proof
      let x be object;
      assume x in InvertibleOperators(X,Y); then
      consider u be Point of S such that
      A2: x = u & u is invertible;
      take y = Inv u;
      y is invertible by A2,LM60;
      hence y in InvertibleOperators(Y,X);
      thus P1[x,y] by A2;
    end;
    consider I be Function of InvertibleOperators(X,Y),InvertibleOperators(Y,X)
    such that
    A3: for x being object st x in InvertibleOperators(X,Y)
        holds P1[x,I . x] from FUNCT_2:sch 1(A1);
    take I;
    A4: for u be Point of S
        st u in InvertibleOperators(X,Y)
        holds I.u = Inv u
    proof
      let u be Point of S;
      assume u in InvertibleOperators(X,Y); then
      P1[u,I . u] by A3;
      hence I.u = Inv u;
    end;
    A5: InvertibleOperators(X,Y) <> {}
        implies InvertibleOperators(Y,X) <> {}
    proof
      assume InvertibleOperators(X,Y) <> {}; then
      consider x be object such that
      A6: x in InvertibleOperators(X,Y) by XBOOLE_0:def 1;
      consider u be Point of S such that
      A7: x = u & u is invertible by A6;
      Inv u is invertible by A7,LM60; then
      Inv u in InvertibleOperators(Y,X);
      hence InvertibleOperators(Y,X) <> {};
    end;
    B7: for x1, x2 being object
        st x1 in InvertibleOperators(X,Y)
         & x2 in InvertibleOperators(X,Y)
         & I . x1 = I . x2
    holds x1 = x2
    proof
      let x1, x2 be object;
      assume that
      A8: x1 in InvertibleOperators(X,Y)
        & x2 in InvertibleOperators(X,Y) and
      A9: I . x1 = I . x2;
      reconsider u1 = x1, u2 = x2 as Point of S by A8;
      A10: ( ex v1 be Point of S st u1=v1 & v1 is invertible )
         & ( ex v2 be Point of S st u2=v2 & v2 is invertible ) by A8;
      A11: I.u1 = Inv u1 by A4,A8;
      u1 = Inv Inv u1 by A10,LM60
        .= Inv Inv u2 by A4,A8,A9,A11
        .= u2 by A10,LM60;
      hence x1 = x2;
    end;
    now
      let y be object;
      assume y in InvertibleOperators(Y,X); then
      consider v be Point of R_NormSpace_of_BoundedLinearOperators(Y,X)
      such that
      A15: y = v & v is invertible;
      Inv v is invertible by A15,LM60; then
      A16: Inv v in InvertibleOperators(X,Y);
      Inv Inv v = v by A15,LM60;
      then y = I.(Inv v) by A4,A15,A16;
      hence y in rng I by A5,A16,FUNCT_2:4;
    end; then
    InvertibleOperators(Y,X) c= rng I by TARSKI:def 3;
    hence thesis by A4,A5,B7,FUNCT_2:19,XBOOLE_0:def 10;
  end;
