let X be RealUnitarySpace; :: thesis: for seq being sequence of X
for n being Nat holds ||.((Partial_Sums seq) . n).|| <= (Partial_Sums ||.seq.||) . n
let seq be sequence of X; :: thesis: for n being Nat holds ||.((Partial_Sums seq) . n).|| <= (Partial_Sums ||.seq.||) . n
defpred S₁[ Nat] means ||.((Partial_Sums seq) . $1).|| <= (Partial_Sums ||.seq.||) . $1;

A1: now :: thesis: for n being Nat st S₁[n] holds
S₁[n + 1]

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume S₁[n] ; :: thesis: S₁[n + 1]
then A2: ||.((Partial_Sums seq) . n).|| + ||.(seq . (n + 1)).|| <= ((Partial_Sums ||.seq.||) . n) + ||.(seq . (n + 1)).|| by XREAL_1:6;
(Partial_Sums seq) . (n + 1) = ((Partial_Sums seq) . n) + (seq . (n + 1)) by Def1;
then ||.((Partial_Sums seq) . (n + 1)).|| <= ||.((Partial_Sums seq) . n).|| + ||.(seq . (n + 1)).|| by BHSP_1:30;
then ||.((Partial_Sums seq) . (n + 1)).|| <= ((Partial_Sums ||.seq.||) . n) + ||.(seq . (n + 1)).|| by A2, XXREAL_0:2;
then ||.((Partial_Sums seq) . (n + 1)).|| <= ((Partial_Sums ||.seq.||) . n) + (||.seq.|| . (n + 1)) by BHSP_2:def 3;
hence S₁[n + 1] by SERIES_1:def 1; :: thesis: verum

end;

(Partial_Sums ||.seq.||) . 0 = ||.seq.|| . 0 by SERIES_1:def 1
.= ||.(seq . 0).|| by BHSP_2:def 3 ;
then A3: S₁[ 0 ] by Def1;
thus for n being Nat holds S₁[n] from NAT_1:sch 2(A3, A1); :: thesis: verum