let S be non empty set ; :: thesis:
let T be Subset of S; :: thesis:
let A be free ECIW-strict preIfWhileAlgebra; :: thesis:
let g be Function of [:S,(ElementaryInstructions A):],S; :: thesis:
consider s0 being Element of S;
let F be Function of (Funcs S,S),(Funcs S,S); :: thesis:
assume A1: for h being Element of Funcs S,S holds (F . h) * h = F . h ; :: thesis:
set Z = Funcs S,S;
deffunc H₁( Element of A) -> set = (curry' g) . $1;
A2: for I being Element of A st I in ElementaryInstructions A holds
H₁(I) in Funcs S,S

proof

let I be Element of A; :: thesis:
assume A3: I in ElementaryInstructions A ; :: thesis:
then reconsider B = ElementaryInstructions A as non empty set ;
reconsider I = I as Element of B by A3;
reconsider g = g as Function of [:S,B:],S ;
(curry' g) . I is Element of Funcs S,S ;
hence H₁(I) in Funcs S,S ; :: thesis:

end;

reconsider idS = id S as Element of Funcs S,S by ALTCAT_1:2;
deffunc H₂( Element of Funcs S,S, Element of Funcs S,S) -> Element of Funcs S,S = $2 * $1;
deffunc H₃( Element of Funcs S,S, Element of Funcs S,S, Element of Funcs S,S) -> Element of Funcs S,S = ($2,T +* $3) * $1;
deffunc H₄( Element of Funcs S,S, Element of Funcs S,S) -> Element of Funcs S,S = (T,(F . ($1 * $2)) iter ($1 * $2)) * $1;
consider h being Function of the carrier of A,(Funcs S,S) such that
A4: for I being Element of A st I in ElementaryInstructions A holds
h . I = H₁(I) and
A5: h . (EmptyIns A) = idS and
A6: for I1, I2 being Element of A holds h . (I1 \; I2) = H₂(h . I1,h . I2) and
A7: for C, I1, I2 being Element of A holds h . (if-then-else C,I1,I2) = H₃(h . C,h . I1,h . I2) and
A8: for C, I being Element of A holds h . (while C,I) = H₄(h . C,h . I) from AOFA_000:sch 7(A2);
h in Funcs the carrier of A,(Funcs S,S) by FUNCT_2:11;
then uncurry' h in Funcs [:S,the carrier of A:],S by FUNCT_6:20;
then reconsider f = uncurry' h as Function of [:S,the carrier of A:],S by FUNCT_2:121;
A9: ( dom h = the carrier of A & dom idS = S ) by FUNCT_2:def 1;
A10: f is complying_with_empty-instruction

proof

let s be Element of S; :: according to AOFA_000:def 28 :: thesis:
idS . s = s by FUNCT_1:35;
hence f . s,(EmptyIns A) = s by A5, A9, FUNCT_5:46; :: thesis:

end;

A11: f is complying_with_catenation

proof

let s be Element of S; :: according to AOFA_000:def 29 :: thesis:
let I1, I2 be Element of A; :: thesis:
A12: ( dom (h . (I1 \; I2)) = S & dom (h . I1) = S & dom (h . I2) = S ) by FUNCT_2:def 1;
hence f . s,(I1 \; I2) = (h . (I1 \; I2)) . s by A9, FUNCT_5:46
.= ((h . I2) * (h . I1)) . s by A6
.= (h . I2) . ((h . I1) . s) by FUNCT_2:21
.= (h . I2) . (f . s,I1) by A9, A12, FUNCT_5:46
.= f . (f . s,I1),I2 by A9, A12, FUNCT_5:46 ;
:: thesis:

end;

A13: f complies_with_if_wrt T

proof

let s be Element of S; :: according to AOFA_000:def 30 :: thesis:
let C, I1, I2 be Element of A; :: thesis:
A14: ( dom (h . (if-then-else C,I1,I2)) = S & dom (h . C) = S & dom (h . I1) = S & dom (h . I2) = S ) by FUNCT_2:def 1;
then A15: f . s,(if-then-else C,I1,I2) = (h . (if-then-else C,I1,I2)) . s by A9, FUNCT_5:46
.= H₃(h . C,h . I1,h . I2) . s by A7
.= ((h . I1),T +* (h . I2)) . ((h . C) . s) by FUNCT_2:21 ;
A16: f . s,C = (h . C) . s by A9, A14, FUNCT_5:46;

hereby :: thesis:

assume f . s,C in T ; :: thesis:
hence f . s,(if-then-else C,I1,I2) = (h . I1) . (f . s,C) by A14, A15, A16, Th4
.= f . (f . s,C),I1 by A9, A14, FUNCT_5:46 ;
:: thesis:

end;

assume f . s,C nin T ; :: thesis:
hence f . s,(if-then-else C,I1,I2) = (h . I2) . (f . s,C) by A14, A15, A16, Th5
.= f . (f . s,C),I2 by A9, A14, FUNCT_5:46 ;
:: thesis:

end;

f complies_with_while_wrt T

proof

let s be Element of S; :: according to AOFA_000:def 31 :: thesis:
let C, I be Element of A; :: thesis:
A17: ( dom (h . (while C,I)) = S & dom (h . C) = S & dom (h . I) = S & dom ((h . C) * (h . I)) = S ) by FUNCT_2:def 1;
then A18: f . s,(while C,I) = (h . (while C,I)) . s by A9, FUNCT_5:46
.= H₄(h . C,h . I) . s by A8
.= (T,(F . ((h . C) * (h . I))) iter ((h . C) * (h . I))) . ((h . C) . s) by FUNCT_2:21 ;
A19: f . s,C = (h . C) . s by A9, A17, FUNCT_5:46;
A20: rng ((h . C) * (h . I)) c= S ;
A21: (F . ((h . C) * (h . I))) * ((h . C) * (h . I)) = F . ((h . C) * (h . I)) by A1;

hereby :: thesis:

assume f . s,C in T ; :: thesis:
hence f . s,(while C,I) = (T,(F . ((h . C) * (h . I))) iter ((h . C) * (h . I))) . (((h . C) * (h . I)) . (f . s,C)) by A17, A18, A19, A20, A21, Th11
.= (T,(F . ((h . C) * (h . I))) iter ((h . C) * (h . I))) . ((h . C) . ((h . I) . (f . s,C))) by FUNCT_2:21
.= ((T,(F . ((h . C) * (h . I))) iter ((h . C) * (h . I))) * (h . C)) . ((h . I) . (f . s,C)) by FUNCT_2:21
.= ((T,(F . ((h . C) * (h . I))) iter ((h . C) * (h . I))) * (h . C)) . (f . (f . s,C),I) by A9, A17, FUNCT_5:46
.= (h . (while C,I)) . (f . (f . s,C),I) by A8
.= f . (f . (f . s,C),I),(while C,I) by A9, A17, FUNCT_5:46 ;
:: thesis:

end;

thus ( f . s,C nin T implies f . s,(while C,I) = f . s,C ) by A17, A18, A19, A20, Th12; :: thesis:

end;

then reconsider f = f as ExecutionFunction of A,S,T by A10, A11, A13, Def32;
take f ; :: thesis:
dom f = [:S,the carrier of A:] by FUNCT_2:def 1;
then [:S,(ElementaryInstructions A):] c= dom f by ZFMISC_1:118;
then A22: ( dom g = [:S,(ElementaryInstructions A):] & (dom f) /\ [:S,(ElementaryInstructions A):] = [:S,(ElementaryInstructions A):] ) by FUNCT_2:def 1, XBOOLE_1:28;

now

let a be set ; :: thesis:
assume A23: a in dom g ; :: thesis:
then consider s, I being set such that
A24: ( s in S & I in ElementaryInstructions A & a = [s,I] ) by ZFMISC_1:def 2;
reconsider s = s as Element of S by A24;
reconsider I = I as Element of A by A24;
reconsider EI = (curry' g) . I as Element of Funcs S,S by A24, A2;
g in Funcs [:S,(ElementaryInstructions A):],S by FUNCT_2:11;
then curry' g in Funcs (ElementaryInstructions A),(Funcs S,S) by A23, FUNCT_6:19;
then A25: ( dom (curry' g) = ElementaryInstructions A & dom EI = S & dom (h . I) = S ) by FUNCT_2:169;
thus g . a = g . s,I by A24
.= EI . s by A24, A25, FUNCT_5:41
.= (h . I) . s by A24, A4
.= f . s,I by A9, A25, FUNCT_5:46
.= f . a by A24 ; :: thesis:

end;

hence f | [:S,(ElementaryInstructions A):] = g by A22, FUNCT_1:68; :: thesis:
let C, I be Element of A; :: thesis:
let s be Element of S; :: thesis:
assume A26: not f iteration_terminates_for I \; C,f . s,C ; :: thesis:
A27: ( dom (h . (while C,I)) = S & dom (h . C) = S & dom (h . I) = S & dom ((h . C) * (h . I)) = S ) by FUNCT_2:def 1;
then A28: f . s,(while C,I) = (h . (while C,I)) . s by A9, FUNCT_5:46
.= H₄(h . C,h . I) . s by A8
.= (T,(F . ((h . C) * (h . I))) iter ((h . C) * (h . I))) . ((h . C) . s) by FUNCT_2:21 ;
A29: f . s,C = (h . C) . s by A9, A27, FUNCT_5:46;
A30: rng ((h . C) * (h . I)) c= S ;
rng h c= Funcs S,S ;
then h = curry' f by FUNCT_5:55;
then A31: (curry' f) . (I \; C) = (h . C) * (h . I) by A6;
then ((h . C) * (h . I)) orbit (f . s,C) c= T by A26, Th87;
hence f . s,(while C,I) = (F . ((curry' f) . (I \; C))) . (f . s,C) by A31, A27, A28, A29, A30, Def7; :: thesis: