let s1, s2 be State of SCM+FSA ; :: thesis:
let I be Program of SCM+FSA ; :: thesis:
assume A1: I +* (Start-At (insloc 0 )) c= s1 ; :: thesis:
assume A2: I is_pseudo-closed_on s1 ; :: thesis:
let n be Element of NAT ; :: thesis:
assume A3: ProgramPart (Relocated I,n) c= s2 ; :: thesis:
defpred S₁[ Element of NAT ] means ( $1 <= pseudo-LifeSpan s1,I implies ( (IC (Computation s1,$1)) + n = IC (Computation s2,$1) & DataPart (Computation s1,$1) = DataPart (Computation s2,$1) ) );
assume A4: IC s2 = insloc n ; :: thesis:
assume A5: DataPart s1 = DataPart s2 ; :: thesis:

hereby :: thesis:

defpred S₂[ Element of NAT ] means ( $1 < pseudo-LifeSpan s1,I implies ( (IC (Computation s1,$1)) + n = IC (Computation s2,$1) & IncAddr (CurInstr (Computation s1,$1)),n = CurInstr (Computation s2,$1) & DataPart (Computation s1,$1) = DataPart (Computation s2,$1) ) );
let i be Element of NAT ; :: thesis:
assume A7: i < pseudo-LifeSpan s1,I ; :: thesis:
A8: for k being Element of NAT st S₂[k] holds
S₂[k + 1]

proof

A9: I c= I +* (Start-At (insloc 0 )) by SCMFSA8A:9;
then A10: dom I c= dom (I +* (Start-At (insloc 0 ))) by GRFUNC_1:8;
let k be Element of NAT ; :: thesis:
assume A11: S₂[k] ; :: thesis:
reconsider l = IC (Computation s1,(k + 1)) as Element of NAT by ORDINAL1:def 13;
reconsider j = CurInstr (Computation s1,(k + 1)) as Instruction of SCM+FSA ;
assume A12: k + 1 < pseudo-LifeSpan s1,I ; :: thesis:
A13: IC (Computation s2,(k + 1)) in NAT by AMI_1:def 4;
A14: Computation s1,(k + 1) = Following (Computation s1,k) by AMI_1:14
.= Exec (CurInstr (Computation s1,k)),(Computation s1,k) ;
s1 +* (I +* (Start-At (insloc 0 ))) = s1 by A1, FUNCT_4:79;
then A15: IC (Computation s1,(k + 1)) in dom I by A2, A12, SCMFSA8A:def 5;
dom (ProgramPart I) = (dom I) /\ NAT by RELAT_1:90;
then A16: l in dom (ProgramPart I) by A15, XBOOLE_0:def 4;
A17: Computation s2,(k + 1) = Following (Computation s2,k) by AMI_1:14
.= Exec (CurInstr (Computation s2,k)),(Computation s2,k) ;
A18: k + 0 < k + 1 by XREAL_1:8;
hence A19: (IC (Computation s1,(k + 1))) + n = IC (Computation s2,(k + 1)) by A11, A12, A14, A17, SCMFSA6A:41, XXREAL_0:2; :: thesis:
then IC (Computation s2,(k + 1)) in dom (Relocated I,n) by A15, SCMFSA_5:4;
then IC (Computation s2,(k + 1)) in (dom (Relocated I,n)) /\ NAT by A13, XBOOLE_0:def 4;
then A20: IC (Computation s2,(k + 1)) in dom (ProgramPart (Relocated I,n)) by RELAT_1:90;
j = s1 . (IC (Computation s1,(k + 1))) by AMI_1:54
.= (I +* (Start-At (insloc 0 ))) . (IC (Computation s1,(k + 1))) by A1, A10, A15, GRFUNC_1:8
.= I . l by A9, A15, GRFUNC_1:8 ;
hence IncAddr (CurInstr (Computation s1,(k + 1))),n = (Relocated I,n) . (l + n) by A16, SCMFSA_5:7
.= (ProgramPart (Relocated I,n)) . (IC (Computation s2,(k + 1))) by A19, FUNCT_1:72
.= s2 . (IC (Computation s2,(k + 1))) by A3, A20, GRFUNC_1:8
.= CurInstr (Computation s2,(k + 1)) by AMI_1:54 ;
:: thesis:
thus DataPart (Computation s1,(k + 1)) = DataPart (Computation s2,(k + 1)) by A11, A12, A18, A14, A17, SCMFSA6A:41, XXREAL_0:2; :: thesis:

end;

A21: S₂[ 0 ]

proof

A22: IC (Computation (s1 +* (I +* (Start-At (insloc 0 )))),0 ) = IC (s1 +* (I +* (Start-At (insloc 0 )))) by AMI_1:13
.= IC ((s1 +* I) +* (Start-At (insloc 0 ))) by FUNCT_4:15
.= insloc 0 by AMI_1:111 ;
assume 0 < pseudo-LifeSpan s1,I ; :: thesis:
then A23: insloc 0 in dom I by A2, A22, SCMFSA8A:def 5;
then A24: insloc 0 in dom (ProgramPart I) by AMI_1:105;
A25: IC SCM+FSA in dom (I +* (Start-At (insloc 0 ))) by SF_MASTR:65;
IC (Computation s1,0 ) = s1 . (IC SCM+FSA ) by AMI_1:13
.= (I +* (Start-At (insloc 0 ))) . (IC SCM+FSA ) by A1, A25, GRFUNC_1:8
.= insloc 0 by SF_MASTR:66 ;
hence (IC (Computation s1,0 )) + n = IC (Computation s2,0 ) by A4, AMI_1:13; :: thesis:
A26: I c= I +* (Start-At (insloc 0 )) by SCMFSA8A:9;
then A27: dom I c= dom (I +* (Start-At (insloc 0 ))) by GRFUNC_1:8;
(insloc 0 ) + n in dom (Relocated I,n) by A23, SCMFSA_5:4;
then A28: insloc (0 + n) in dom (ProgramPart (Relocated I,n)) by AMI_1:106;
IC SCM+FSA in dom (I +* (Start-At (insloc 0 ))) by SF_MASTR:65;
then A29: s1 . (IC s1) = s1 . ((I +* (Start-At (insloc 0 ))) . (IC SCM+FSA )) by A1, GRFUNC_1:8
.= s1 . (insloc 0 ) by SF_MASTR:66
.= (I +* (Start-At (insloc 0 ))) . (insloc 0 ) by A1, A27, A23, GRFUNC_1:8
.= I . (insloc 0 ) by A26, A23, GRFUNC_1:8 ;
thus IncAddr (CurInstr (Computation s1,0 )),n = IncAddr (CurInstr s1),n by AMI_1:13
.= (Relocated I,n) . (insloc (0 + n)) by A29, A24, SCMFSA_5:7
.= (ProgramPart (Relocated I,n)) . (insloc n) by FUNCT_1:72
.= CurInstr s2 by A3, A4, A28, GRFUNC_1:8
.= CurInstr (Computation s2,0 ) by AMI_1:13 ; :: thesis:
thus DataPart (Computation s1,0 ) = DataPart s2 by A5, AMI_1:13
.= DataPart (Computation s2,0 ) by AMI_1:13 ; :: thesis:

end;

for k being Element of NAT holds S₂[k] from NAT_1:sch 1(A21, A8);
hence IncAddr (CurInstr (Computation s1,i)),n = CurInstr (Computation s2,i) by A7; :: thesis:

end;

A30: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis:
assume A31: S₁[k] ; :: thesis:
set i = CurInstr (Computation s1,k);
A32: Computation s2,(k + 1) = Following (Computation s2,k) by AMI_1:14
.= Exec (CurInstr (Computation s2,k)),(Computation s2,k) ;
assume A33: k + 1 <= pseudo-LifeSpan s1,I ; :: thesis:
then A34: k + 1 <= (pseudo-LifeSpan s1,I) + 1 by NAT_1:12;
A35: k < pseudo-LifeSpan s1,I by A33, NAT_1:13;
A36: Computation s1,(k + 1) = Following (Computation s1,k) by AMI_1:14
.= Exec (CurInstr (Computation s1,k)),(Computation s1,k) ;
hence (IC (Computation s1,(k + 1))) + n = IC (Exec (IncAddr (CurInstr (Computation s1,k)),n),(Computation s2,k)) by A31, A34, SCMFSA6A:41, XREAL_1:8
.= IC (Computation s2,(k + 1)) by A6, A35, A32 ;
:: thesis:
thus DataPart (Computation s1,(k + 1)) = DataPart (Exec (IncAddr (CurInstr (Computation s1,k)),n),(Computation s2,k)) by A31, A34, A36, SCMFSA6A:41, XREAL_1:8
.= DataPart (Computation s2,(k + 1)) by A6, A35, A32 ; :: thesis:

end;

let i be Element of NAT ; :: thesis:
assume A37: i <= pseudo-LifeSpan s1,I ; :: thesis:
A38: S₁[ 0 ]

proof

assume 0 <= pseudo-LifeSpan s1,I ; :: thesis:
A39: IC SCM+FSA in dom (I +* (Start-At (insloc 0 ))) by SF_MASTR:65;
IC (Computation s1,0 ) = s1 . (IC SCM+FSA ) by AMI_1:13
.= (I +* (Start-At (insloc 0 ))) . (IC SCM+FSA ) by A1, A39, GRFUNC_1:8
.= insloc 0 by SF_MASTR:66 ;
hence (IC (Computation s1,0 )) + n = IC (Computation s2,0 ) by A4, AMI_1:13; :: thesis:
thus DataPart (Computation s1,0 ) = DataPart s2 by A5, AMI_1:13
.= DataPart (Computation s2,0 ) by AMI_1:13 ; :: thesis:

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A38, A30);
hence ( (IC (Computation s1,i)) + n = IC (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) ) by A37; :: thesis: