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

theorem Th25:
  Sum ((X-powers (1/2))^\(n+1)) <= (1/2)|^n
proof
  set r = 1/2;
  |.r.| = r by ABSVALUE:def 1;
  then
A1: (r GeoSeq)^\(n+1) is summable by SERIES_1:12,24;
A2: now
    let m be Nat;
A3: ((X-powers r)^\(n+1)).m = (X-powers r).(m+(n+1)) by NAT_1:def 3;
A4: m+(n+1) in X & (X-powers r).(m+(n+1)) = r|^(m+(n+1)) or not m+(n+1) in
    X & (X-powers r).(m+(n+1)) = 0 by Def5;
    hence 0 <= ((X-powers r)^\(n+1)).m by A3,PREPOWER:6;
A5: (r GeoSeq).(m+(n+1)) = r|^(m+(n+1)) by PREPOWER:def 1;
    ((r GeoSeq)^\(n+1)).m = (r GeoSeq).(m+(n+1)) by NAT_1:def 3;
    hence ((X-powers r)^\(n+1)).m <= ((r GeoSeq)^\(n+1)).m by A5,A3,A4,
PREPOWER:6;
  end;
  Sum (((1/2) GeoSeq)^\(n+1)) = (1/2)|^n by Th23;
  hence thesis by A1,A2,SERIES_1:20;
end;
