let s1, s2 be State of SCM+FSA ; :: thesis:
let J be parahalting Program of SCM+FSA ; :: thesis:
assume A1: J +* (Start-At 0 ,SCM+FSA ) c= s1 ; :: thesis:
set JAt = J +* (Start-At 0 ,SCM+FSA );
A2: 0 in dom J by AFINSQ_1:69;
dom J misses dom (Start-At 0 ,SCM+FSA ) by SF_MASTR:64;
then A3: J c= J +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:33;
then A4: dom J c= dom (J +* (Start-At 0 ,SCM+FSA )) by GRFUNC_1:8;
A5: IC SCM+FSA in dom (J +* (Start-At 0 ,SCM+FSA )) by SF_MASTR:65;
then A6: s1 . (IC s1) = s1 . ((J +* (Start-At 0 ,SCM+FSA )) . (IC SCM+FSA )) by A1, GRFUNC_1:8
.= s1 . 0 by SF_MASTR:66
.= (J +* (Start-At 0 ,SCM+FSA )) . 0 by A1, A4, A2, GRFUNC_1:8
.= J . 0 by A3, A2, GRFUNC_1:8 ;
A7: IC (Comput (ProgramPart s1),s1,0 ) = s1 . (IC SCM+FSA ) by AMI_1:13
.= (J +* (Start-At 0 ,SCM+FSA )) . (IC SCM+FSA ) by A1, A5, GRFUNC_1:8
.= 0 by SF_MASTR:66 ;
ProgramPart J = J by RELAT_1:209;
then A8: 0 in dom (ProgramPart J) by AFINSQ_1:69;
let n be Element of NAT ; :: thesis:
assume that
A9: ProgramPart (Relocated J,n) c= s2 and
A10: IC s2 = n and
A11: DataPart s1 = DataPart s2 ; :: thesis:
A12: DataPart (Comput (ProgramPart s1),s1,0 ) = DataPart s2 by A11, AMI_1:13
.= DataPart (Comput (ProgramPart s2),s2,0 ) by AMI_1:13 ;
defpred S₁[ Nat] means ( (IC (Comput (ProgramPart s1),s1,$1)) + n = IC (Comput (ProgramPart s2),s2,$1) & IncAddr (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,$1)),(Comput (ProgramPart s1),s1,$1)),n = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,$1)),(Comput (ProgramPart s2),s2,$1) & DataPart (Comput (ProgramPart s1),s1,$1) = DataPart (Comput (ProgramPart s2),s2,$1) );
A13: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

dom J misses dom (Start-At 0 ,SCM+FSA ) by SF_MASTR:64;
then A14: J c= J +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:33;
then A15: dom J c= dom (J +* (Start-At 0 ,SCM+FSA )) by GRFUNC_1:8;
let k be Element of NAT ; :: thesis:
T: ProgramPart s1 = ProgramPart (Comput (ProgramPart s1),s1,k) by AMI_1:123;
A16: Comput (ProgramPart s1),s1,(k + 1) = Following (ProgramPart s1),(Comput (ProgramPart s1),s1,k) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k) by T ;
reconsider l = IC (Comput (ProgramPart s1),s1,(k + 1)) as Element of NAT ;
reconsider j = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,(k + 1))),(Comput (ProgramPart s1),s1,(k + 1)) as Instruction of SCM+FSA ;
T: ProgramPart s2 = ProgramPart (Comput (ProgramPart s2),s2,k) by AMI_1:123;
A17: Comput (ProgramPart s2),s2,(k + 1) = Following (ProgramPart s2),(Comput (ProgramPart s2),s2,k) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart s2),s2,k)),(Comput (ProgramPart s2),s2,k)),(Comput (ProgramPart s2),s2,k) by T ;
A18: IC (Comput (ProgramPart s1),s1,(k + 1)) in dom J by A1, Def2;
assume A19: S₁[k] ; :: thesis:
hence A20: (IC (Comput (ProgramPart s1),s1,(k + 1))) + n = IC (Comput (ProgramPart s2),s2,(k + 1)) by A16, A17, SCMFSA6A:41; :: thesis:
then ( IC (Comput (ProgramPart s2),s2,(k + 1)) in NAT & IC (Comput (ProgramPart s2),s2,(k + 1)) in dom (Relocated J,n) ) by A18, AMISTD_2:71;
then IC (Comput (ProgramPart s2),s2,(k + 1)) in (dom (Relocated J,n)) /\ NAT by XBOOLE_0:def 4;
then A21: IC (Comput (ProgramPart s2),s2,(k + 1)) in dom (ProgramPart (Relocated J,n)) by RELAT_1:90;
dom (ProgramPart J) = (dom J) /\ NAT by RELAT_1:90;
then A22: l in dom (ProgramPart J) by A18, XBOOLE_0:def 4;
Y: (ProgramPart (Comput (ProgramPart s1),s1,(k + 1))) /. (IC (Comput (ProgramPart s1),s1,(k + 1))) = (Comput (ProgramPart s1),s1,(k + 1)) . (IC (Comput (ProgramPart s1),s1,(k + 1))) by COMPOS_1:38;
Z: (ProgramPart (Comput (ProgramPart s2),s2,(k + 1))) /. (IC (Comput (ProgramPart s2),s2,(k + 1))) = (Comput (ProgramPart s2),s2,(k + 1)) . (IC (Comput (ProgramPart s2),s2,(k + 1))) by COMPOS_1:38;
j = s1 . (IC (Comput (ProgramPart s1),s1,(k + 1))) by Y, AMI_1:54
.= (J +* (Start-At 0 ,SCM+FSA )) . (IC (Comput (ProgramPart s1),s1,(k + 1))) by A1, A15, A18, GRFUNC_1:8
.= J . l by A14, A18, GRFUNC_1:8 ;
hence IncAddr (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,(k + 1))),(Comput (ProgramPart s1),s1,(k + 1))),n = (Relocated J,n) . (l + n) by A22, AMISTD_2:76
.= (ProgramPart (Relocated J,n)) . (IC (Comput (ProgramPart s2),s2,(k + 1))) by A20, FUNCT_1:72
.= s2 . (IC (Comput (ProgramPart s2),s2,(k + 1))) by A9, A21, GRFUNC_1:8
.= CurInstr (ProgramPart (Comput (ProgramPart s2),s2,(k + 1))),(Comput (ProgramPart s2),s2,(k + 1)) by Z, AMI_1:54 ;
:: thesis:
thus DataPart (Comput (ProgramPart s1),s1,(k + 1)) = DataPart (Comput (ProgramPart s2),s2,(k + 1)) by A19, A16, A17, SCMFSA6A:41; :: thesis:

end;

let i be Element of NAT ; :: thesis:
0 in dom J by AFINSQ_1:69;
then 0 + n in dom (Relocated J,n) by AMISTD_2:71;
then A23: 0 + n in dom (ProgramPart (Relocated J,n)) by COMPOS_1:16;
u: Comput (ProgramPart s1),s1,0 = s1 by AMI_1:13;
v: Comput (ProgramPart s2),s2,0 = s2 by AMI_1:13;
Z: (ProgramPart s1) /. (IC s1) = s1 . (IC s1) by COMPOS_1:38;
V: (ProgramPart s2) /. (IC s2) = s2 . (IC s2) by COMPOS_1:38;
IncAddr (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,0 )),(Comput (ProgramPart s1),s1,0 )),n = IncAddr (CurInstr (ProgramPart s1),s1),n by u
.= (Relocated J,n) . (0 + n) by A6, A8, Z, AMISTD_2:76
.= (ProgramPart (Relocated J,n)) . n by FUNCT_1:72
.= CurInstr (ProgramPart s2),s2 by A9, A10, A23, V, GRFUNC_1:8
.= CurInstr (ProgramPart (Comput (ProgramPart s2),s2,0 )),(Comput (ProgramPart s2),s2,0 ) by v ;
then A24: S₁[ 0 ] by A10, A7, A12, AMI_1:13;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A24, A13);
hence ( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & IncAddr (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i)),n = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) ) ; :: thesis: