let a be real number ; :: thesis: for s being Real_Sequence st ( for n being Element of NAT st n >= 1 holds
( s . n = a & s . 0 = 1 ) ) holds
for n being Element of NAT st n >= 1 holds
(Partial_Product s) . n = a |^ n
let s be Real_Sequence; :: thesis: ( ( for n being Element of NAT st n >= 1 holds
( s . n = a & s . 0 = 1 ) ) implies for n being Element of NAT st n >= 1 holds
(Partial_Product s) . n = a |^ n )
defpred S₁[ Nat] means (Partial_Product s) . $1 = a |^ $1;
assume A1: for n being Element of NAT st n >= 1 holds
( s . n = a & s . 0 = 1 ) ; :: thesis: for n being Element of NAT st n >= 1 holds
(Partial_Product s) . 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 in NAT by ORDINAL1:def 13;
then (Partial_Product s) . (n + 1) = ((Partial_Product s) . n) * (s . (n + 1)) by SERIES_3:def 1
.= (a |^ n) * a by A1, A3, A4
.= a |^ (n + 1) by NEWTON:11 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Product s) . (1 + 0) = ((Partial_Product s) . 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 by NEWTON:10 ;
then A5: S₁[1] ;
for n being Nat st n >= 1 holds
S₁[n] from NAT_1:sch 8(A5, A2);
hence for n being Element of NAT st n >= 1 holds
(Partial_Product s) . n = a |^ n ; :: thesis: verum