reserve a,b,c for set;

theorem Th21:
  for X being set, r being Real st 0 < r & r < 1 holds X
  -powers r is summable
proof
  let X be set, r be Real such that
A1: 0 < r and
A2: r < 1;
A3: now
    let n be Nat;
    n in X & (X-powers r).n = r|^n or not n in X & (X-powers r).n = 0 by Def5;
    hence 0 <= (X-powers r).n by A1,PREPOWER:6;
  end;
A4: now
     reconsider m = 1 as Nat;
    take m;
    let n be Nat such that
    m <= n;
A5: (r GeoSeq).n = r|^n by PREPOWER:def 1;
    n in X & (X-powers r).n = r|^n or not n in X & (X-powers r).n = 0 by Def5;
    hence (X-powers r).n <= (r GeoSeq).n by A1,A5,PREPOWER:6;
  end;
  |.r.| = r by A1,ABSVALUE:def 1;
  then r GeoSeq is summable by A2,SERIES_1:24;
  hence thesis by A4,A3,SERIES_1:19;
end;
