let seq be ExtREAL_sequence; :: thesis: ( ( for n being Nat holds 0 < seq . n ) implies for m being Nat holds 0 < (Partial_Sums seq) . m )
defpred S₁[ Nat] means 0 < (Partial_Sums seq) . $1;
assume A1: for n being Nat holds 0 < seq . n ; :: thesis: for m being Nat holds 0 < (Partial_Sums seq) . m
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
A4: (Partial_Sums seq) . (k + 1) = ((Partial_Sums seq) . k) + (seq . (k + 1)) by Def1;
seq . (k + 1) > 0 by A1;
hence S₁[k + 1] by A3, A4; :: thesis: verum

end;

(Partial_Sums seq) . 0 = seq . 0 by Def1;
then A5: S₁[ 0 ] by A1;
thus for m being Nat holds S₁[m] from NAT_1:sch 2(A5, A2); :: thesis: verum