let s be Real_Sequence; :: thesis: ( ( for n being Nat st n >= 1 holds
( s . n = n & s . 0 = 1 ) ) implies for n being Nat st n >= 1 holds
(Partial_Product s) . n = n ! )
defpred S₁[ Nat] means (Partial_Product s) . $1 = $1 ! ;
assume A1: for n being Nat st n >= 1 holds
( s . n = n & s . 0 = 1 ) ; :: thesis: for n being Nat st n >= 1 holds
(Partial_Product s) . n = 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 = n ! ; :: thesis: S₁[n + 1]
A4: n + 1 >= 1 by NAT_1:11;
(Partial_Product s) . (n + 1) = ((Partial_Product s) . n) * (s . (n + 1)) by SERIES_3:def 1
.= (n !) * (n + 1) by A1, A3, A4
.= (n + 1) ! by NEWTON:15 ;
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 ! by A1, NEWTON:13 ;
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 Nat st n >= 1 holds
(Partial_Product s) . n = n ! ; :: thesis: verum