let a be Real; :: thesis: for s being Real_Sequence st ( for n being Nat st n >= 1 holds
( s . n = a & s . 0 = 1 ) ) holds
for n being Nat st n >= 1 holds
() . n = a |^ n

let s be Real_Sequence; :: thesis: ( ( for n being Nat st n >= 1 holds
( s . n = a & s . 0 = 1 ) ) implies for n being Nat st n >= 1 holds
() . n = a |^ n )

defpred S1[ Nat] means () . \$1 = a |^ \$1;
assume A1: for n being Nat st n >= 1 holds
( s . n = a & s . 0 = 1 ) ; :: thesis: for n being Nat st n >= 1 holds
() . n = a |^ n

A2: for n being Nat st n >= 1 & S1[n] holds
S1[n + 1]
proof
let n be Nat; :: thesis: ( n >= 1 & S1[n] implies S1[n + 1] )
assume that
n >= 1 and
A3: (Partial_Product s) . n = a |^ n ; :: thesis: S1[n + 1]
A4: n + 1 >= 1 by NAT_1:11;
() . (n + 1) = (() . n) * (s . (n + 1)) by SERIES_3:def 1
.= (a |^ n) * a by A1, A3, A4
.= a |^ (n + 1) by NEWTON:6 ;
hence S1[n + 1] ; :: thesis: verum
end;
() . (1 + 0) = () * (s . (1 + 0)) by SERIES_3:def 1
.= (s . 0) * (s . (1 + 0)) by SERIES_3:def 1
.= 1 * (s . 1) by A1
.= 1 * a by A1
.= a |^ 1 ;
then A5: S1 ;
for n being Nat st n >= 1 holds
S1[n] from NAT_1:sch 8(A5, A2);
hence for n being Nat st n >= 1 holds
() . n = a |^ n ; :: thesis: verum