let s be State of SCM+FSA; :: thesis: for I being paraclosed Program of SCM+FSA
for J being 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 +* (I ';' J))),(s +* (I ';' J)),m) equal_outside NAT
let I be paraclosed Program of SCM+FSA; :: thesis: for J being 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 +* (I ';' J))),(s +* (I ';' J)),m) equal_outside NAT
let J be 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 +* (I ';' J))),(s +* (I ';' J)),m) equal_outside NAT )
assume that
A1: I +* (Start-At (0,SCM+FSA)) c= s and
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 +* (I ';' J))),(s +* (I ';' J)),m) equal_outside NAT
defpred S₁[ Nat] means ( $1 <= LifeSpan ((ProgramPart s),s) implies Comput ((ProgramPart s),s,$1), Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),$1) equal_outside NAT );
A3: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

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 +* (I ';' J))),(s +* (I ';' J)),m) equal_outside NAT ) ; :: thesis: S₁[m + 1]
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: ( I ';' J c= Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m) & dom I c= dom (I ';' J) ) by AMI_1:81, FUNCT_4:26, XBOOLE_1:7;
T: ProgramPart s = ProgramPart (Comput ((ProgramPart s),s,m)) by AMI_1:123;
A6: 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 ;
T: ProgramPart (s +* (I ';' J)) = ProgramPart (Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m)) by AMI_1:123;
A7: Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),(m + 1)) = Following ((ProgramPart (s +* (I ';' J))),(Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m))),(Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m)))),(Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m))) by T ;
A8: IC (Comput ((ProgramPart s),s,m)) in dom I by A1, Def2;
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 A9: CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,m))) = I . (IC (Comput ((ProgramPart s),s,m))) by A8, Y, TX, GRFUNC_1:8;
assume A10: m + 1 <= LifeSpan ((ProgramPart s),s) ; :: thesis: Comput ((ProgramPart s),s,(m + 1)), Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),(m + 1)) equal_outside NAT
then A11: IC (Comput ((ProgramPart s),s,m)) = IC (Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m)) by A4, COMPOS_1:24, NAT_1:13;
Y: (ProgramPart (Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m))) /. (IC (Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m))) = (Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m)) . (IC (Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m))) by COMPOS_1:38;
m < LifeSpan ((ProgramPart s),s) by A10, NAT_1:13;
then I . (IC (Comput ((ProgramPart s),s,m))) <> halt SCM+FSA by A2, A9, EXTPRO_1:def 14;
then CurInstr ((ProgramPart (Comput ((ProgramPart s),s,m))),(Comput ((ProgramPart s),s,m))) = (I ';' J) . (IC (Comput ((ProgramPart s),s,m))) by A8, A9, TX, SCMFSA6A:54
.= CurInstr ((ProgramPart (Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m))),(Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m))) by A11, A8, A5, Y, GRFUNC_1:8 ;
hence Comput ((ProgramPart s),s,(m + 1)), Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),(m + 1)) equal_outside NAT by A4, A10, A6, A7, NAT_1:13, SCMFSA6A:32; :: thesis: verum

end;

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