let n be Nat; :: thesis: for s being Real_Sequence st ( for n being Nat holds s . n = 1 / ((2 -Root (n + 1)) + (2 -Root n)) ) holds
() . n = 2 -Root (n + 1)

let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = 1 / ((2 -Root (n + 1)) + (2 -Root n)) ) implies () . n = 2 -Root (n + 1) )
defpred S1[ Nat] means () . \$1 = 2 -Root (\$1 + 1);
assume A1: for n being Nat holds s . n = 1 / ((2 -Root (n + 1)) + (2 -Root n)) ; :: thesis: () . n = 2 -Root (n + 1)
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 (Partial_Sums s) . n = 2 -Root (n + 1) ; :: thesis: S1[n + 1]
then () . (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;
() . 0 = s . 0 by SERIES_1:def 1
.= 1 / ((2 -Root (0 + 1)) + ()) by A1
.= 1 / ((2 -Root (0 + 1)) + 0) by PREPOWER:def 2
.= 1 / 1 by PREPOWER:20
.= 2 -Root (0 + 1) by PREPOWER:20 ;
then A3: S1[ 0 ] ;
for n being Nat holds S1[n] from NAT_1:sch 2(A3, A2);
hence (Partial_Sums s) . n = 2 -Root (n + 1) ; :: thesis: verum