let s be State of SCM+FSA; :: thesis: for I being paraclosed Program of SCM+FSA st ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) halts_on s +* (I +* (Start-At (0,SCM+FSA))) holds
for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) holds
Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k) equal_outside NAT
set SA0 = Start-At (0,SCM+FSA);
let I be paraclosed Program of SCM+FSA; :: thesis: ( ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) halts_on s +* (I +* (Start-At (0,SCM+FSA))) implies for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) holds
Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k) equal_outside NAT )
assume A1: ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) halts_on s +* (I +* (Start-At (0,SCM+FSA))) ; :: thesis: for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) holds
Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k) equal_outside NAT
set s1 = s +* (I +* (Start-At (0,SCM+FSA)));
let J be Program of SCM+FSA; :: thesis: for k being Element of NAT st k <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) holds
Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),k) equal_outside NAT
A2: I +* (Start-At (0,SCM+FSA)) c= s +* (I +* (Start-At (0,SCM+FSA))) by FUNCT_4:26;
set s2 = s +* ((I ';' J) +* (Start-At (0,SCM+FSA)));
defpred S₁[ Nat] means ( $1 <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) implies Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),$1), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),$1) equal_outside NAT );
A3: (I ';' J) +* (Start-At (0,SCM+FSA)) c= s +* ((I ';' J) +* (Start-At (0,SCM+FSA))) by FUNCT_4:26;
A4: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

dom (I ';' J) = (dom (Directed I)) \/ (dom (ProgramPart (Relocated (J,(card I))))) by FUNCT_4:def 1
.= (dom I) \/ (dom (ProgramPart (Relocated (J,(card I))))) by FUNCT_4:105 ;
then A5: dom I c= dom (I ';' J) by XBOOLE_1:7;
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A6: ( m <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) implies Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),m) equal_outside NAT ) ; :: thesis: S₁[m + 1]
dom (I ';' J) misses dom (Start-At (0,SCM+FSA)) by COMPOS_1:140;
then I ';' J c= (I ';' J) +* (Start-At (0,SCM+FSA)) by FUNCT_4:33;
then I ';' J c= s +* ((I ';' J) +* (Start-At (0,SCM+FSA))) by A3, XBOOLE_1:1;
then A7: I ';' J c= Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),m) by AMI_1:81;
T: ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m)) by AMI_1:123;
A8: Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(m + 1)) = Following ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m)))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) by T ;
T: ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),m)) by AMI_1:123;
A9: Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),(m + 1)) = Following ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),m)))),(Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),m))) by T ;
A10: IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m)) in dom I by A2, Def2;
Y: (ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) /. (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) = (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m)) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) by COMPOS_1:38;
TX: ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),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 +* (I +* (Start-At (0,SCM+FSA))) by A2, XBOOLE_1:1;
then I c= Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m) by AMI_1:81;
then A11: CurInstr ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) = I . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) by A10, Y, TX, GRFUNC_1:8;
assume A12: m + 1 <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) ; :: thesis: Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(m + 1)), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),(m + 1)) equal_outside NAT
then A13: IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m)) = IC (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),m)) by A6, COMPOS_1:24, NAT_1:13;
Y: (ProgramPart (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),m))) /. (IC (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),m))) = (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),m)) . (IC (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),m))) by COMPOS_1:38;
m < LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) by A12, NAT_1:13;
then I . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) <> halt SCM+FSA by A1, A11, EXTPRO_1:def 14;
then CurInstr ((ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) = (I ';' J) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) by A10, A11, TX, SCMFSA6A:54
.= CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),m))) by A13, A10, A7, A5, Y, GRFUNC_1:8 ;
hence Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(m + 1)), Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),(m + 1)) equal_outside NAT by A6, A12, A8, A9, NAT_1:13, SCMFSA6A:32; :: thesis: verum

end;

A14: ( Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),0) = s +* (I +* (Start-At (0,SCM+FSA))) & Comput ((ProgramPart (s +* ((I ';' J) +* (Start-At (0,SCM+FSA))))),(s +* ((I ';' J) +* (Start-At (0,SCM+FSA)))),0) = s +* ((I ';' J) +* (Start-At (0,SCM+FSA))) ) by EXTPRO_1:3;
A15: ( (s +* (Start-At (0,SCM+FSA))) +* I,s +* (Start-At (0,SCM+FSA)) equal_outside NAT & s +* (Start-At (0,SCM+FSA)),(s +* (Start-At (0,SCM+FSA))) +* (I ';' J) equal_outside NAT ) by FUNCT_7:28, FUNCT_7:132;
A16: s +* ((I ';' J) +* (Start-At (0,SCM+FSA))) = Initialize (s +* (I ';' J)) by FUNCT_4:15
.= (Initialize s) +* (I ';' J) by COMPOS_1:83 ;
s +* (I +* (Start-At (0,SCM+FSA))) = Initialize (s +* I) by FUNCT_4:15
.= (Initialize s) +* I by COMPOS_1:83 ;
then A17: S₁[ 0 ] by A16, A15, A14, FUNCT_7:29;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A17, A4); :: thesis: verum