let s be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds s . n = n ) implies for n being Element of NAT holds (Partial_Sums s) . n = (n * (n + 1)) / 2 )
assume A1: for n being Element of NAT holds s . n = n ; :: thesis: for n being Element of NAT holds (Partial_Sums s) . n = (n * (n + 1)) / 2
defpred S₁[ Element of NAT ] means (Partial_Sums s) . $1 = ($1 * ($1 + 1)) / 2;
(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= (0 * (0 + 1)) / 2 by A1 ;
then A2: S₁[ 0 ] ;
A3: 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 (Partial_Sums s) . n = (n * (n + 1)) / 2 ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = ((n * (n + 1)) / 2) + (s . (n + 1)) by SERIES_1:def 1
.= ((n * (n + 1)) / 2) + (n + 1) by A1
.= ((n * (n + 1)) + ((n + 1) * 2)) / 2 ;
hence S₁[n + 1] ; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A2, A3);
hence for n being Element of NAT holds (Partial_Sums s) . n = (n * (n + 1)) / 2 ; :: thesis: verum