let s be State of SCM+FSA; :: thesis: for I being paraclosed Program of SCM+FSA st I +* (Start-At (0,SCM+FSA)) c= s & ProgramPart s halts_on s holds
for m being Element of NAT st m <= LifeSpan ((ProgramPart s),s) holds
Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m) equal_outside NAT
let I be paraclosed Program of SCM+FSA; :: thesis: ( I +* (Start-At (0,SCM+FSA)) c= s & ProgramPart s halts_on s implies for m being Element of NAT st m <= LifeSpan ((ProgramPart s),s) holds
Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m) equal_outside NAT )
assume A1: I +* (Start-At (0,SCM+FSA)) c= s ; :: thesis: ( not ProgramPart s halts_on s or for m being Element of NAT st m <= LifeSpan ((ProgramPart s),s) holds
Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m) equal_outside NAT )
defpred S₁[ Nat] means ( $1 <= LifeSpan ((ProgramPart s),s) implies Comput ((ProgramPart s),s,$1), Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),$1) equal_outside NAT );
assume A2: ProgramPart s halts_on s ; :: thesis: for m being Element of NAT st m <= LifeSpan ((ProgramPart s),s) holds
Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m) equal_outside NAT
A3: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

set sI = s +* (loop I);
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A4: ( m <= LifeSpan ((ProgramPart s),s) implies Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m) equal_outside NAT ) ; :: thesis: S₁[m + 1]
A5: IC (Comput ((ProgramPart s),s,m)) in dom I by A1, SCMFSA6B:def 2;
then A6: IC (Comput ((ProgramPart s),s,m)) in dom (loop I) by FUNCT_4:105;
Y: (ProgramPart (Comput ((ProgramPart s),s,m))) /. (IC (Comput ((ProgramPart s),s,m))) = (Comput ((ProgramPart s),s,m)) . (IC (Comput ((ProgramPart s),s,m))) by COMPOS_1:38;
TX: ProgramPart s = ProgramPart (Comput ((ProgramPart s),s,m)) by AMI_1:123;
dom I misses dom (Start-At (0,SCM+FSA)) by COMPOS_1:140;
then I c= I +* (Start-At (0,SCM+FSA)) by FUNCT_4:33;
then I c= s by A1, XBOOLE_1:1;
then I c= Comput ((ProgramPart s),s,m) by AMI_1:81;
then A7: CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,m))) = I . (IC (Comput ((ProgramPart s),s,m))) by A5, Y, TX, GRFUNC_1:8;
A8: loop I c= Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m) by AMI_1:81, FUNCT_4:26;
T: ProgramPart (s +* (loop I)) = ProgramPart (Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m)) by AMI_1:123;
A9: Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),(m + 1)) = Following ((ProgramPart (s +* (loop I))),(Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m))),(Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m)))),(Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m))) by T ;
T: ProgramPart s = ProgramPart (Comput ((ProgramPart s),s,m)) by AMI_1:123;
A10: Comput ((ProgramPart s),s,(m + 1)) = Following ((ProgramPart s),(Comput ((ProgramPart s),s,m))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s),s,m))),(Comput ((ProgramPart s),s,m)))),(Comput ((ProgramPart s),s,m))) by T ;
Y: (ProgramPart (Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m))) /. (IC (Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m))) = (Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m)) . (IC (Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m))) by COMPOS_1:38;
assume A11: m + 1 <= LifeSpan ((ProgramPart s),s) ; :: thesis: Comput ((ProgramPart s),s,(m + 1)), Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),(m + 1)) equal_outside NAT
then m < LifeSpan ((ProgramPart s),s) by NAT_1:13;
then I . (IC (Comput ((ProgramPart s),s,m))) <> halt SCM+FSA by A2, A7, EXTPRO_1:def 14;
then CurInstr ((ProgramPart (Comput ((ProgramPart s),s,m))),(Comput ((ProgramPart s),s,m))) = (loop I) . (IC (Comput ((ProgramPart s),s,m))) by A7, TX, FUNCT_4:111
.= (Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m)) . (IC (Comput ((ProgramPart s),s,m))) by A8, A6, GRFUNC_1:8
.= CurInstr ((ProgramPart (Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m))),(Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),m))) by A4, A11, Y, COMPOS_1:24, NAT_1:13 ;
hence Comput ((ProgramPart s),s,(m + 1)), Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),(m + 1)) equal_outside NAT by A4, A11, A10, A9, NAT_1:13, SCMFSA6A:32; :: thesis: verum

end;

A12: Comput ((ProgramPart (s +* (loop I))),(s +* (loop I)),0) = s +* (loop I) by EXTPRO_1:3;
Comput ((ProgramPart s),s,0) = s by EXTPRO_1:3;
then A13: S₁[ 0 ] by A12, FUNCT_7:132;
thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A13, A3); :: thesis: verum