let s be State of SCM+FSA ; :: thesis: for I, J being Program of SCM+FSA st I is_closed_on s & 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, J be Program of SCM+FSA ; :: thesis: ( I is_closed_on s & 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 is_closed_on s and
A2: I +* (Start-At 0 ,SCM+FSA ) c= s and
A3: 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[ Element of 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 );
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;
set sIJ = s +* (I ';' J);
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A6: ( 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]
A7: I ';' J c= Comput (ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m by AMI_1:81, FUNCT_4:26;
T: ProgramPart (s +* (I ';' J)) = ProgramPart (Comput (ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m) by AMI_1:123;
A8: 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 ;
T: ProgramPart s = ProgramPart (Comput (ProgramPart s),s,m) by AMI_1:123;
A9: 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 ;
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;
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 A6, COMPOS_1:24, NAT_1:13;
s = s +* (I +* (Start-At 0 ,SCM+FSA )) by A2, FUNCT_4:79;
then A12: IC (Comput (ProgramPart s),s,m) in dom I by A1, SCMFSA7B:def 7;
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 A2, XBOOLE_1:1;
then I c= Comput (ProgramPart s),s,m by AMI_1:81;
then A13: CurInstr (ProgramPart s),(Comput (ProgramPart s),s,m) = I . (IC (Comput (ProgramPart s),s,m)) by A12, Y, TX, GRFUNC_1:8;
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 A3, A13, 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 A12, A13, 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, A12, A7, 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 A6, A10, A9, A8, NAT_1:13, SCMFSA6A:32; :: thesis: verum

end;

A14: Comput (ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),0 = s +* (I ';' J) by AMI_1:13;
Comput (ProgramPart s),s,0 = s by AMI_1:13;
then A15: S₁[ 0 ] by A14, FUNCT_7:132;
thus for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A15, A4); :: thesis: verum