let n be Element of NAT ; :: thesis: for s being Real_Sequence st ( for n being Element of NAT holds s . n = 1 / ((2 -Root (n + 1)) + (2 -Root n)) ) holds
(Partial_Sums s) . n = 2 -Root (n + 1)
let s be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds s . n = 1 / ((2 -Root (n + 1)) + (2 -Root n)) ) implies (Partial_Sums s) . n = 2 -Root (n + 1) )
defpred S₁[ Element of NAT ] means (Partial_Sums s) . $1 = 2 -Root ($1 + 1);
assume A1: for n being Element of NAT holds s . n = 1 / ((2 -Root (n + 1)) + (2 -Root n)) ; :: thesis: (Partial_Sums s) . n = 2 -Root (n + 1)
A2: 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 = 2 -Root (n + 1) ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = (2 -Root (n + 1)) + (s . (n + 1)) by SERIES_1:def 1
.= (2 -Root (n + 1)) + (1 / ((2 -Root ((n + 1) + 1)) + (2 -Root (n + 1)))) by A1
.= (2 -Root (n + 1)) + ((2 -Root (n + 2)) - (2 -Root (n + 1))) by Lm7
.= 2 -Root (n + 2) ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= 1 / ((2 -Root (0 + 1)) + (2 -Root 0 )) by A1
.= 1 / ((2 -Root (0 + 1)) + 0 ) by PREPOWER:def 3
.= 1 / 1 by PREPOWER:29
.= 2 -Root (0 + 1) by PREPOWER:29 ;
then A3: S₁[ 0 ] ;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A3, A2);
hence (Partial_Sums s) . n = 2 -Root (n + 1) ; :: thesis: verum