reserve X,Y,Z for non trivial RealBanachSpace;

theorem LM2:
  for S be Banach_Algebra, w be Point of S st ||.w.|| < 1
  holds 1.S + w is invertible
      & (-w) GeoSeq is norm_summable
      & (1.S + w) " = Sum ( (-w) GeoSeq )
      & ||.(1.S + w) ".|| <= 1 / ( 1 - ||.w.|| )
  proof
    let S be Banach_Algebra,
        w be Point of S;
    assume
    A1: ||.w.|| < 1;
    set x = 1.S + w;
    A2: ||.(1. S) - x.|| = ||.x -1.S .|| by NORMSP_1:7
    .= ||.w.|| by RLVECT_4:1; then
    A3: x is invertible & x" = Sum ( (1. S - x) GeoSeq ) by A1,LOPBAN_3:42;
    1.S - x = 1.S - 1.S - w by RLVECT_1:27
    .= 0.S - w by RLVECT_1:15
    .= -w by RLVECT_1:14;
    hence thesis by A1,A2,A3,LM1;
  end;
