let s, s1, s2 be Real_Sequence; :: thesis: ( s = s1 (#) s2 & ( for n being Element of NAT holds
( s1 . n >= 0 & s2 . n >= 0 ) ) implies for n being Element of NAT holds (Partial_Sums s) . n <= ((Partial_Sums s1) . n) * ((Partial_Sums s2) . n) )
assume A1: ( s = s1 (#) s2 & ( for n being Element of NAT holds
( s1 . n >= 0 & s2 . n >= 0 ) ) ) ; :: thesis: for n being Element of NAT holds (Partial_Sums s) . n <= ((Partial_Sums s1) . n) * ((Partial_Sums s2) . n)
defpred S₁[ Element of NAT ] means (Partial_Sums s) . $1 <= ((Partial_Sums s1) . $1) * ((Partial_Sums s2) . $1);
A2: (Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= (s1 . 0 ) * (s2 . 0 ) by A1, SEQ_1:12 ;
((Partial_Sums s1) . 0 ) * ((Partial_Sums s2) . 0 ) = (s1 . 0 ) * ((Partial_Sums s2) . 0 ) by SERIES_1:def 1
.= (s1 . 0 ) * (s2 . 0 ) by SERIES_1:def 1 ;
then A3: S₁[ 0 ] by A2;
A4: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof

let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A5: (Partial_Sums s) . n <= ((Partial_Sums s1) . n) * ((Partial_Sums s2) . n) ; :: thesis: S₁[n + 1]
A6: (Partial_Sums s) . (n + 1) = ((Partial_Sums s) . n) + (s . (n + 1)) by SERIES_1:def 1
.= ((Partial_Sums s) . n) + ((s1 . (n + 1)) * (s2 . (n + 1))) by A1, SEQ_1:12 ;
set j = (Partial_Sums s) . n;
set u = (Partial_Sums s1) . n;
set v = (Partial_Sums s2) . n;
set w = s1 . (n + 1);
set h = s2 . (n + 1);
A7: ((Partial_Sums s1) . (n + 1)) * ((Partial_Sums s2) . (n + 1)) = (((Partial_Sums s1) . n) + (s1 . (n + 1))) * ((Partial_Sums s2) . (n + 1)) by SERIES_1:def 1
.= (((Partial_Sums s1) . n) + (s1 . (n + 1))) * (((Partial_Sums s2) . n) + (s2 . (n + 1))) by SERIES_1:def 1
.= (((((Partial_Sums s1) . n) * ((Partial_Sums s2) . n)) + (((Partial_Sums s1) . n) * (s2 . (n + 1)))) + ((s1 . (n + 1)) * ((Partial_Sums s2) . n))) + ((s1 . (n + 1)) * (s2 . (n + 1))) ;
A8: ((Partial_Sums s) . n) + ((s1 . (n + 1)) * (s2 . (n + 1))) <= (((Partial_Sums s1) . n) * ((Partial_Sums s2) . n)) + ((s1 . (n + 1)) * (s2 . (n + 1))) by A5, XREAL_1:8;
A9: ( (Partial_Sums s1) . n >= 0 & (Partial_Sums s2) . n >= 0 ) by A1, Th34;
( s1 . (n + 1) >= 0 & s2 . (n + 1) >= 0 ) by A1;
then ( ((Partial_Sums s1) . n) * (s2 . (n + 1)) >= 0 & (s1 . (n + 1)) * ((Partial_Sums s2) . n) >= 0 ) by A9;
then (((Partial_Sums s1) . n) * (s2 . (n + 1))) + ((s1 . (n + 1)) * ((Partial_Sums s2) . n)) >= 0 ;
then (((Partial_Sums s1) . n) * ((Partial_Sums s2) . n)) + ((s1 . (n + 1)) * (s2 . (n + 1))) <= ((((Partial_Sums s1) . n) * ((Partial_Sums s2) . n)) + ((s1 . (n + 1)) * (s2 . (n + 1)))) + ((((Partial_Sums s1) . n) * (s2 . (n + 1))) + ((s1 . (n + 1)) * ((Partial_Sums s2) . n))) by XREAL_1:33;
hence S₁[n + 1] by A6, A7, A8, XXREAL_0:2; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A3, A4);
hence for n being Element of NAT holds (Partial_Sums s) . n <= ((Partial_Sums s1) . n) * ((Partial_Sums s2) . n) ; :: thesis: verum