let s be State of SCM+FSA ; :: thesis: for I being Program of SCM+FSA st I is_closed_onInit s & I is_halting_onInit s holds
( CurInstr (Computation (s +* (Initialized (loop I))),(LifeSpan (s +* (Initialized I)))) = goto (insloc 0 ) & ( for m being Element of NAT st m <= LifeSpan (s +* (Initialized I)) holds
CurInstr (Computation (s +* (Initialized (loop I))),m) <> halt SCM+FSA ) )
let I be Program of SCM+FSA ; :: thesis: ( I is_closed_onInit s & I is_halting_onInit s implies ( CurInstr (Computation (s +* (Initialized (loop I))),(LifeSpan (s +* (Initialized I)))) = goto (insloc 0 ) & ( for m being Element of NAT st m <= LifeSpan (s +* (Initialized I)) holds
CurInstr (Computation (s +* (Initialized (loop I))),m) <> halt SCM+FSA ) ) )
set s1 = s +* (Initialized I);
set s2 = s +* (Initialized (loop I));
assume A1: ( I is_closed_onInit s & I is_halting_onInit s ) ; :: thesis: ( CurInstr (Computation (s +* (Initialized (loop I))),(LifeSpan (s +* (Initialized I)))) = goto (insloc 0 ) & ( for m being Element of NAT st m <= LifeSpan (s +* (Initialized I)) holds
CurInstr (Computation (s +* (Initialized (loop I))),m) <> halt SCM+FSA ) )
then A2: s +* (Initialized I) is halting by Def5;
set k = LifeSpan (s +* (Initialized I));
A3: CurInstr (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) = halt SCM+FSA by A2, AMI_1:def 46;
A4: IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) = IC (Computation (s +* (Initialized (loop I))),(LifeSpan (s +* (Initialized I)))) by A1, Th68, AMI_1:121;
A5: IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) in dom I by A1, Def4;
then IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) in dom (I +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 )))) by FUNCT_4:13;
then A6: IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) in dom (Initialized I) by FUNCT_4:15;

A7: now

thus CurInstr (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) = (s +* (Initialized I)) . (IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I))))) by AMI_1:54
.= (Initialized I) . (IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I))))) by A6, FUNCT_4:14
.= (I +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 )))) . (IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I))))) by FUNCT_4:15
.= I . (IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I))))) by Th70, FUNCT_4:12 ; :: thesis: verum

end;

dom (loop I) = dom I by FUNCT_4:105;
then IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) in dom ((loop I) +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 )))) by A5, FUNCT_4:13;
then A8: IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) in dom (Initialized (loop I)) by FUNCT_4:15;
X: rng I c= the Instructions of SCM+FSA by AMI_1:118;
y: dom (id the Instructions of SCM+FSA ) = the Instructions of SCM+FSA by RELAT_1:71;
thus A11: CurInstr (Computation (s +* (Initialized (loop I))),(LifeSpan (s +* (Initialized I)))) = (s +* (Initialized (loop I))) . (IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I))))) by A4, AMI_1:54
.= (Initialized (loop I)) . (IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I))))) by A8, FUNCT_4:14
.= ((loop I) +* (((intloc 0 ) .--> 1) +* (Start-At (insloc 0 )))) . (IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I))))) by FUNCT_4:15
.= (loop I) . (IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I))))) by Th70, FUNCT_4:12
.= (((id the Instructions of SCM+FSA ) +* (halt SCM+FSA ),(goto (insloc 0 ))) * I) . (IC (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I))))) by X, FUNCT_7:118
.= ((id the Instructions of SCM+FSA ) +* (halt SCM+FSA ),(goto (insloc 0 ))) . (halt SCM+FSA ) by A3, A5, A7, FUNCT_1:23
.= goto (insloc 0 ) by y, FUNCT_7:33 ; :: thesis: for m being Element of NAT st m <= LifeSpan (s +* (Initialized I)) holds
CurInstr (Computation (s +* (Initialized (loop I))),m) <> halt SCM+FSA
let m be Element of NAT ; :: thesis: ( m <= LifeSpan (s +* (Initialized I)) implies CurInstr (Computation (s +* (Initialized (loop I))),m) <> halt SCM+FSA )
assume A12: m <= LifeSpan (s +* (Initialized I)) ; :: thesis: CurInstr (Computation (s +* (Initialized (loop I))),m) <> halt SCM+FSA

per cases ( m < LifeSpan (s +* (Initialized I)) or m = LifeSpan (s +* (Initialized I)) ) by A12, XXREAL_0:1;

suppose A13: m < LifeSpan (s +* (Initialized I)) ; :: thesis: CurInstr (Computation (s +* (Initialized (loop I))),m) <> halt SCM+FSA

then CurInstr (Computation (s +* (Initialized I)),m) <> halt SCM+FSA by A2, AMI_1:def 46;
hence CurInstr (Computation (s +* (Initialized (loop I))),m) <> halt SCM+FSA by A1, A13, Th69; :: thesis: verum

end;

suppose m = LifeSpan (s +* (Initialized I)) ; :: thesis: CurInstr (Computation (s +* (Initialized (loop I))),m) <> halt SCM+FSA

hence CurInstr (Computation (s +* (Initialized (loop I))),m) <> halt SCM+FSA by A11, SCMFSA_2:47, SCMFSA_2:124; :: thesis: verum

end;

end;