reserve a,b,c for set;
reserve r for Real,
  X for set,
  n for Element of NAT;

theorem
  0 < r & r < 1 implies Sum (X-powers r) <= Sum (r GeoSeq)
proof
  assume that
A1: 0 < r and
A2: r < 1;
A3: now
    let n be Nat;
A4: 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;
    (r GeoSeq).n = r|^n by PREPOWER:def 1;
    hence (X-powers r).n <= (r GeoSeq).n by A1,A4,PREPOWER:6;
  end;
  |.r.| = r by A1,ABSVALUE:def 1;
  then r GeoSeq is summable by A2,SERIES_1:24;
  hence thesis by A3,SERIES_1:20;
end;
