let s1, s2 be Real_Sequence; :: thesis: ( ( for n being Nat holds s1 . n <= s2 . n ) implies for n being Nat holds (Partial_Sums s1) . n <= (Partial_Sums s2) . n )
defpred S₁[ Nat] means (Partial_Sums s1) . $1 <= (Partial_Sums s2) . $1;
assume A1: for n being Nat holds s1 . n <= s2 . n ; :: thesis: for n being Nat holds (Partial_Sums s1) . n <= (Partial_Sums s2) . n
A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A3: (Partial_Sums s1) . n <= (Partial_Sums s2) . n ; :: thesis: S₁[n + 1]
A4: s1 . (n + 1) <= s2 . (n + 1) by A1;
( (Partial_Sums s1) . (n + 1) = ((Partial_Sums s1) . n) + (s1 . (n + 1)) & (Partial_Sums s2) . (n + 1) = ((Partial_Sums s2) . n) + (s2 . (n + 1)) ) by Def1;
hence S₁[n + 1] by A3, A4, XREAL_1:7; :: thesis: verum

end;

( (Partial_Sums s2) . 0 = s2 . 0 & (Partial_Sums s1) . 0 = s1 . 0 ) by Def1;
then A5: S₁[ 0 ] by A1;
thus for n being Nat holds S₁[n] from NAT_1:sch 2(A5, A2); :: thesis: verum