reserve X,Y,Z for non trivial RealBanachSpace;

theorem LM1:
  for X be Banach_Algebra, z be Point of X st ||.z.|| < 1 holds
    z GeoSeq is norm_summable
  & ||.Sum ( z GeoSeq ).|| <= 1 / ( 1 - ||.z.|| )
  proof
    let X be Banach_Algebra;
    let z be Element of X;
    A1: for n being Nat holds
        ( 0 <= ||.(z GeoSeq).|| . n
        & ||.(z GeoSeq).|| . n <= (||.z.|| GeoSeq) . n )
    proof
      defpred S1[Nat] means
      ( 0 <= ||.(z GeoSeq).|| . $1
      & ||.(z GeoSeq).|| . $1 <= (||.z.|| GeoSeq) . $1 );
      A3: for k being Nat st S1[k] holds S1[k + 1]
      proof
        let k be Nat;
        ||.(((z GeoSeq) . k) * z).|| <= ||.((z GeoSeq) . k).|| * ||.z.||
          by LOPBAN_2:def 9; then
        A4: ||.(((z GeoSeq) . k) * z).|| <= (||.(z GeoSeq).|| . k) * ||.z.||
            by NORMSP_0:def 4;
        assume S1[k]; then
        (||.(z GeoSeq).|| . k) * ||.z.|| <= ((||.z.|| GeoSeq) . k) * ||.z.||
          by XREAL_1:64; then
        A5: ||.(((z GeoSeq) . k) * z).|| <= ((||.z.|| GeoSeq) . k) * ||.z.||
          by A4,XXREAL_0:2;
        ||.(z GeoSeq).|| . (k + 1)
         = ||.((z GeoSeq) . (k + 1)).|| by NORMSP_0:def 4
        .= ||.(((z GeoSeq) . k) * z).|| by LOPBAN_3:def 9;
        hence S1[k + 1] by A5,PREPOWER:3;
      end;
      ||.((z GeoSeq) . 0).|| = ||.(1. X).|| by LOPBAN_3:def 9
      .= 1 by LOPBAN_2:def 10
      .= (||.z.|| GeoSeq) . 0 by PREPOWER:3; then
      A6: S1[ 0 ] by NORMSP_0:def 4;
      for n being Nat holds S1[n] from NAT_1:sch 2(A6,A3);
      hence for n being Nat holds
          ( 0 <= ||.(z GeoSeq).|| . n
          & ||.(z GeoSeq).|| . n <= (||.z.|| GeoSeq) . n );
    end;
    assume ||.z.|| < 1; then
    |.||.z.||.| < 1 by ABSVALUE:def 1; then
    A7: ||.z.|| GeoSeq is summable
      & Sum (||.z.|| GeoSeq) = 1 / (1 - ||.z.||) by SERIES_1:24; then
    A8: ||.(z GeoSeq).|| is summable
      & Sum ||.(z GeoSeq).|| <= Sum (||.z.|| GeoSeq) by A1,SERIES_1:20;
    z GeoSeq is norm_summable by A1,A7,SERIES_1:20; then
    ||.Sum (z GeoSeq ).|| <= Sum ||.(z GeoSeq).|| by LM0;
    hence thesis by A7,A8,XXREAL_0:2;
  end;
