let A be Euclidean preIfWhileAlgebra; :: thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0
for n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
for q being Element of Funcs X,INT
for N being natural number st N = q . n holds
(g . q,((s := 1) \; (for-do (i := 2),(i leq n),(i += 1),(s *= i)))) . s = N !
let X be non empty countable set ; :: thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0
for n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
for q being Element of Funcs X,INT
for N being natural number st N = q . n holds
(g . q,((s := 1) \; (for-do (i := 2),(i leq n),(i += 1),(s *= i)))) . s = N !
let b be Element of X; :: thesis: for g being Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0
for n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
for q being Element of Funcs X,INT
for N being natural number st N = q . n holds
(g . q,((s := 1) \; (for-do (i := 2),(i leq n),(i += 1),(s *= i)))) . s = N !
let g be Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0 ; :: thesis: for n, s, i being Variable of g st ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds
for q being Element of Funcs X,INT
for N being natural number st N = q . n holds
(g . q,((s := 1) \; (for-do (i := 2),(i leq n),(i += 1),(s *= i)))) . s = N !
set f = g;
let n, s, i be Variable of g; :: thesis: ( ex d being Function st
( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) implies for q being Element of Funcs X,INT
for N being natural number st N = q . n holds
(g . q,((s := 1) \; (for-do (i := 2),(i leq n),(i += 1),(s *= i)))) . s = N ! )
given d being Function such that A1: d . n = 1 and
A2: d . s = 2 and
A3: d . i = 3 and
A4: d . b = 4 ; :: thesis: for q being Element of Funcs X,INT
for N being natural number st N = q . n holds
(g . q,((s := 1) \; (for-do (i := 2),(i leq n),(i += 1),(s *= i)))) . s = N !
A5: ( n <> i & n <> b & i <> b ) by A1, A3, A4;
set S = Funcs X,INT ;
let q be Element of Funcs X,INT ; :: thesis: for N being natural number st N = q . n holds
(g . q,((s := 1) \; (for-do (i := 2),(i leq n),(i += 1),(s *= i)))) . s = N !
reconsider q1 = g . q,(s := 1) as Element of Funcs X,INT ;
defpred S₁[ Element of Funcs X,INT ] means ex K being natural number st
( K = ($1 . i) - 1 & $1 . s = K ! );
reconsider a = . 2,A,g as INT-Expression of A,g ;
reconsider q2 = g . q1,(i := 2) as Element of Funcs X,INT ;
A6: q2 . i = 2 by Th25;
A7: s <> i by A2, A3;
then q2 . s = q1 . s by Th25;
then A8: S₁[g . q1,(i := a)] by A6, Th25, NEWTON:19;
set I = s *= i;
A9: a . q1 = 2 by FUNCOP_1:13;
A10: s <> b by A2, A4;
A11: for q being Element of Funcs X,INT st S₁[q] holds
( S₁[g . q,((s *= i) \; (i += 1))] & S₁[g . q,(i leq n)] )

proof

let q be Element of Funcs X,INT ; :: thesis: ( S₁[q] implies ( S₁[g . q,((s *= i) \; (i += 1))] & S₁[g . q,(i leq n)] ) )
given Ki being natural number such that A12: Ki = (q . i) - 1 and
A13: q . s = Ki ! ; :: thesis: ( S₁[g . q,((s *= i) \; (i += 1))] & S₁[g . q,(i leq n)] )
reconsider q3 = g . q,(s *= i) as Element of Funcs X,INT ;
reconsider q4 = g . q3,(i += 1) as Element of Funcs X,INT ;
A14: q3 . s = (q . s) * (q . i) by Th33;
q4 . s = q3 . s by A7, Th28;
then A15: q4 . s = (Ki + 1) ! by A12, A13, A14, NEWTON:21;
A16: q4 = g . q,((s *= i) \; (i += 1)) by AOFA_000:def 29;
q4 . i = (q3 . i) + 1 by Th28;
then Ki + 1 = (q4 . i) - 1 by A7, A12, Th33;
hence S₁[g . q,((s *= i) \; (i += 1))] by A16, A15; :: thesis: S₁[g . q,(i leq n)]
reconsider q9 = g . q,(i leq n) as Element of Funcs X,INT ;
A17: q9 . i = q . i by A5, Th35;
q9 . s = q . s by A10, Th35;
hence S₁[g . q,(i leq n)] by A12, A13, A17; :: thesis: verum

end;

reconsider F = for-do (i := a),(i leq n),(i += 1),(s *= i) as Element of A ;
set T = (Funcs X,INT ) \ b,0 ;
let N be natural number ; :: thesis: ( N = q . n implies (g . q,((s := 1) \; (for-do (i := 2),(i leq n),(i += 1),(s *= i)))) . s = N ! )
assume A18: N = q . n ; :: thesis: (g . q,((s := 1) \; (for-do (i := 2),(i leq n),(i += 1),(s *= i)))) . s = N !
A19: F = for-do (i := a),(i leq n),(i += 1),(s *= i) ;
A20: s <> n by A1, A2;
then A21: q1 . n = q . n by Th25;
A22: for q being Element of Funcs X,INT st S₁[q] holds
( (g . q,(s *= i)) . i = q . i & (g . q,(s *= i)) . n = q . n ) by A20, A7, Th32;
A23: ( S₁[g . q1,F] & ( a . q1 <= q1 . n implies (g . q1,F) . i = (q1 . n) + 1 ) & ( a . q1 > q1 . n implies (g . q1,F) . i = a . q1 ) & (g . q1,F) . n = q1 . n ) from AOFA_I00:sch 1(A19, A8, A11, A22, A5);
thus (g . q,((s := 1) \; (for-do (i := 2),(i leq n),(i += 1),(s *= i)))) . s = (g . q1,F) . s by AOFA_000:def 29
.= N ! by A18, A21, A23, A9, NAT_1:27, NEWTON:18, NEWTON:19 ; :: thesis: verum