let s be State of ; :: thesis: for I being paraclosed Program of
for J being Program of st I +* (Start-At (insloc 0 )) c= s & ProgramPart s halts_on s holds
for m being Element of NAT st m <= LifeSpan s holds
Computation s,m, Computation (s +* (I ';' J)),m equal_outside NAT
let I be paraclosed Program of ; :: thesis: for J being Program of st I +* (Start-At (insloc 0 )) c= s & ProgramPart s halts_on s holds
for m being Element of NAT st m <= LifeSpan s holds
Computation s,m, Computation (s +* (I ';' J)),m equal_outside NAT
let J be Program of ; :: thesis: ( I +* (Start-At (insloc 0 )) c= s & ProgramPart s halts_on s implies for m being Element of NAT st m <= LifeSpan s holds
Computation s,m, Computation (s +* (I ';' J)),m equal_outside NAT )
assume that
A1: I +* (Start-At (insloc 0 )) c= s and
A2: ProgramPart s halts_on s ; :: thesis: for m being Element of NAT st m <= LifeSpan s holds
Computation s,m, Computation (s +* (I ';' J)),m equal_outside NAT
defpred S₁[ Element of NAT ] means ( $1 <= LifeSpan s implies Computation s,$1, Computation (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 Computation s,m, Computation (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= Computation (s +* (I ';' J)),m & dom I c= dom (I ';' J) ) by AMI_1:81, FUNCT_4:26, XBOOLE_1:7;
A6: Computation s,(m + 1) = Following (Computation s,m) by AMI_1:14
.= Exec (CurInstr (Computation s,m)),(Computation s,m) ;
A7: Computation (s +* (I ';' J)),(m + 1) = Following (Computation (s +* (I ';' J)),m) by AMI_1:14
.= Exec (CurInstr (Computation (s +* (I ';' J)),m)),(Computation (s +* (I ';' J)),m) ;
A8: IC (Computation s,m) in dom I by A1, Def2;
dom I misses dom (Start-At (insloc 0 )) by SF_MASTR:64;
then I c= I +* (Start-At (insloc 0 )) by FUNCT_4:33;
then I c= s by A1, XBOOLE_1:1;
then I c= Computation s,m by AMI_1:81;
then A9: CurInstr (Computation s,m) = I . (IC (Computation s,m)) by A8, GRFUNC_1:8;
assume A10: m + 1 <= LifeSpan s ; :: thesis: Computation s,(m + 1), Computation (s +* (I ';' J)),(m + 1) equal_outside NAT
then A11: IC (Computation s,m) = IC (Computation (s +* (I ';' J)),m) by A4, AMI_1:121, NAT_1:13;
m < LifeSpan s by A10, NAT_1:13;
then I . (IC (Computation s,m)) <> halt SCM+FSA by A2, A9, AMI_1:def 46;
then CurInstr (Computation s,m) = (I ';' J) . (IC (Computation s,m)) by A8, A9, SCMFSA6A:54
.= CurInstr (Computation (s +* (I ';' J)),m) by A11, A8, A5, GRFUNC_1:8 ;
hence Computation s,(m + 1), Computation (s +* (I ';' J)),(m + 1) equal_outside NAT by A4, A10, A6, A7, NAT_1:13, SCMFSA6A:32; :: thesis: verum

end;

( Computation s,0 = s & Computation (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