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;
TIJ: 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 EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (s +* (I ';' J))),(Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m)))),(Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m))) ;
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 EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,m)))),(Comput ((ProgramPart s),s,m))) ;
Y: (ProgramPart s) /. (IC (Comput ((ProgramPart s),s,m))) = (Comput ((ProgramPart s),s,m)) . (IC (Comput ((ProgramPart s),s,m))) by T, COMPOS_1:38;
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 COMPOS_1:140;
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, GRFUNC_1:8;
Y: (ProgramPart (s +* (I ';' J))) /. (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 TIJ, 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, EXTPRO_1:def 14;
then CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,m))) = (I ';' J) . (IC (Comput ((ProgramPart s),s,m))) by A12, A13, SCMFSA6A:54
.= CurInstr ((ProgramPart (s +* (I ';' J))),(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 EXTPRO_1:3;
Comput ((ProgramPart s),s,0) = s by EXTPRO_1:3;
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