reserve X,Y,Z for non trivial RealBanachSpace;

theorem LM4:
  for X be non trivial RealBanachSpace,
      v be Point of R_NormSpace_of_BoundedLinearOperators(X,X),
      w be Point of R_Normed_Algebra_of_BoundedLinearOperators X
  st v = w
  holds
    ( v is invertible iff w is invertible )
  & ( w is invertible implies v" = w" )
  proof
    let X be non trivial RealBanachSpace,
        v be Point of R_NormSpace_of_BoundedLinearOperators(X,X),
        w be Point of R_Normed_Algebra_of_BoundedLinearOperators X;
    set S = R_Normed_Algebra_of_BoundedLinearOperators X;
    assume
    A1: v = w;
    A2: v is Lipschitzian LinearOperator of X,X by LOPBAN_1:def 9;
    A4: v is invertible implies w is invertible
    proof
      assume
      A5: v is invertible; then
      reconsider x = v" as Point of S;
      A7: v" is Lipschitzian LinearOperator of X,X by A5,LOPBAN_1:def 9;
      reconsider zx = x, zv = v as
        Element of BoundedLinearOperators (X,X);
      dom v = the carrier of X & rng v = the carrier of X
          by A2,A5,FUNCT_2:def 1; then
      A9: v" * v = id X
        & v * v" = id X by A5,FUNCT_1:39;
      A10: v" = modetrans(v",X,X)
        & v = modetrans(v,X,X) by A2,A7,LOPBAN_1:29; then
      A11: v" * v = zx * zv
      .= x * w by A1,LOPBAN_2:def 4;
      v * v" = zv * zx by A10
      .= w * x by A1,LOPBAN_2:def 4;
      hence w is invertible by A9,A11;
    end;
    w is invertible implies v is invertible & v" = w"
    proof
      assume
      A13: w is invertible;
      A14: v is Lipschitzian LinearOperator of X,X by LOPBAN_1:def 9;
      reconsider y = w" as
        Lipschitzian LinearOperator of X,X by LOPBAN_1:def 9;
      reconsider zy = y, zw = w as
        Element of BoundedLinearOperators(X,X);
      A15: y = modetrans(y,X,X) & v = modetrans(v,X,X)
        by A14,LOPBAN_1:29; then
      A16: y * v = zy * zw by A1
      .= w" * w by LOPBAN_2:def 4
      .= id X by A13,LOPBAN_3:def 8;
      A17: v * y = zw * zy by A1,A15
      .= w * w" by LOPBAN_2:def 4
      .= id X by A13,LOPBAN_3:def 8;
      reconsider y0 = y, v0 = v as
        Function of the carrier of X,the carrier of X by LOPBAN_1:def 9;
      A18: dom v = the carrier of X by A14,FUNCT_2:def 1;
      A19: v0 is one-to-one & v0 is onto by A16,A17,FUNCT_2:23; then
      dom y = rng v by FUNCT_2:def 1;
      hence thesis by A16,A18,A19,FUNCT_1:41;
    end;
    hence thesis by A4;
  end;
