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 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 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 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 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 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 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_1:144;
A6: Comput (ProgramPart s),s,(m + 1) = Following (ProgramPart s),(Comput (ProgramPart s),s,m) by AMI_1:14
.= 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_1:144;
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 AMI_1:14
.= 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 AMI_1:150;
dom I misses dom (Start-At 0 ,SCM+FSA ) by SF_MASTR:64;
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 (Comput (ProgramPart s),s,m)),(Comput (ProgramPart s),s,m) = I . (IC (Comput (ProgramPart s),s,m)) by A8, GRFUNC_1:8, Y;
assume A10: m + 1 <= LifeSpan 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, AMI_1:121, 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 AMI_1:150;
m < LifeSpan s by A10, NAT_1:13;
then I . (IC (Comput (ProgramPart s),s,m)) <> halt SCM+FSA by A2, A9, AMI_1:def 46;
then CurInstr (ProgramPart (Comput (ProgramPart s),s,m)),(Comput (ProgramPart s),s,m) = (I ';' J) . (IC (Comput (ProgramPart s),s,m)) by A8, A9, 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, GRFUNC_1:8, Y ;
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 AMI_1:13;
then A12: S₁[ 0 ] by AMI_1:120;
thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A12, A3); :: thesis: verum