let s1, s2 be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA st I is_closed_on s1 & I +* (Start-At (0,SCM+FSA)) c= s1 holds
for n being Element of NAT st ProgramPart (Relocated (I,n)) c= s2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (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)) )
let J be Program of SCM+FSA; :: thesis: ( J is_closed_on s1 & J +* (Start-At (0,SCM+FSA)) c= s1 implies for n being Element of NAT st ProgramPart (Relocated (J,n)) c= s2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (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)) ) )
set JAt = J +* (Start-At (0,SCM+FSA));
assume A1: J is_closed_on s1 ; :: thesis: ( not J +* (Start-At (0,SCM+FSA)) c= s1 or for n being Element of NAT st ProgramPart (Relocated (J,n)) c= s2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (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)) ) )
then A2: 0 in dom J by Th39;
dom J misses dom (Start-At (0,SCM+FSA)) by COMPOS_1:140;
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;
assume A5: J +* (Start-At (0,SCM+FSA)) c= s1 ; :: thesis: for n being Element of NAT st ProgramPart (Relocated (J,n)) c= s2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (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)) )
A6: IC SCM+FSA in dom (J +* (Start-At (0,SCM+FSA))) by COMPOS_1:141;
then A7: s1 . (IC s1) = s1 . (IC (J +* (Start-At (0,SCM+FSA)))) by A5, GRFUNC_1:8
.= s1 . 0 by COMPOS_1:142
.= (J +* (Start-At (0,SCM+FSA))) . 0 by A5, A4, A2, GRFUNC_1:8
.= J . 0 by A3, A2, GRFUNC_1:8 ;
A8: IC (Comput ((ProgramPart s1),s1,0)) = s1 . (IC SCM+FSA) by EXTPRO_1:3
.= IC (J +* (Start-At (0,SCM+FSA))) by A5, A6, GRFUNC_1:8
.= 0 by COMPOS_1:142 ;
ProgramPart J = J by RELAT_1:209;
then A9: 0 in dom (ProgramPart J) by A1, Th39;
let n be Element of NAT ; :: thesis: ( ProgramPart (Relocated (J,n)) c= s2 & IC s2 = n & DataPart s1 = DataPart s2 implies for i being Element of NAT holds
( (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)) ) )
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)) );
assume that
A10: ProgramPart (Relocated (J,n)) c= s2 and
A11: IC s2 = n and
A12: DataPart s1 = DataPart s2 ; :: thesis: for i being Element of NAT holds
( (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)) )
let i be Element of NAT ; :: thesis: ( (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)) )
A13: DataPart (Comput ((ProgramPart s1),s1,0)) = DataPart s2 by A12, EXTPRO_1:3
.= DataPart (Comput ((ProgramPart s2),s2,0)) by EXTPRO_1:3 ;
A14: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

dom J misses dom (Start-At (0,SCM+FSA)) by COMPOS_1:140;
then A15: J c= J +* (Start-At (0,SCM+FSA)) by FUNCT_4:33;
then A16: dom J c= dom (J +* (Start-At (0,SCM+FSA))) by GRFUNC_1:8;
let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
T: ProgramPart s1 = ProgramPart (Comput ((ProgramPart s1),s1,k)) by AMI_1:123;
A17: Comput ((ProgramPart s1),s1,(k + 1)) = Following ((ProgramPart s1),(Comput ((ProgramPart s1),s1,k))) by EXTPRO_1:4
.= 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 ;
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;
T: ProgramPart s2 = ProgramPart (Comput ((ProgramPart s2),s2,k)) by AMI_1:123;
A18: Comput ((ProgramPart s2),s2,(k + 1)) = Following ((ProgramPart s2),(Comput ((ProgramPart s2),s2,k))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,k))),(Comput ((ProgramPart s2),s2,k)))),(Comput ((ProgramPart s2),s2,k))) by T ;
s1 +* (J +* (Start-At (0,SCM+FSA))) = s1 by A5, FUNCT_4:79;
then A20: IC (Comput ((ProgramPart s1),s1,(k + 1))) in dom J by A1, SCMFSA7B:def 7;
assume A21: S₁[k] ; :: thesis: S₁[k + 1]
hence A22: (IC (Comput ((ProgramPart s1),s1,(k + 1)))) + n = IC (Comput ((ProgramPart s2),s2,(k + 1))) by A17, A18, SCMFSA6A:41; :: thesis: ( IncAddr ((CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,(k + 1)))),(Comput ((ProgramPart s1),s1,(k + 1))))),n) = CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,(k + 1)))),(Comput ((ProgramPart s2),s2,(k + 1)))) & DataPart (Comput ((ProgramPart s1),s1,(k + 1))) = DataPart (Comput ((ProgramPart s2),s2,(k + 1))) )
then IC (Comput ((ProgramPart s2),s2,(k + 1))) in dom (Relocated (J,n)) by A20, COMPOS_1:118;
then IC (Comput ((ProgramPart s2),s2,(k + 1))) in (dom (Relocated (J,n))) /\ NAT by XBOOLE_0:def 4;
then A23: 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 A24: l in dom (ProgramPart J) by A20, XBOOLE_0:def 4;
U: (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 A5, A16, A20, GRFUNC_1:8
.= J . l by A15, A20, 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 A24, COMPOS_1:122
.= (ProgramPart (Relocated (J,n))) . (IC (Comput ((ProgramPart s2),s2,(k + 1)))) by A22, FUNCT_1:72
.= s2 . (IC (Comput ((ProgramPart s2),s2,(k + 1)))) by A10, A23, GRFUNC_1:8
.= CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,(k + 1)))),(Comput ((ProgramPart s2),s2,(k + 1)))) by U, AMI_1:54 ;
:: thesis: DataPart (Comput ((ProgramPart s1),s1,(k + 1))) = DataPart (Comput ((ProgramPart s2),s2,(k + 1)))
thus DataPart (Comput ((ProgramPart s1),s1,(k + 1))) = DataPart (Comput ((ProgramPart s2),s2,(k + 1))) by A21, A17, A18, SCMFSA6A:41; :: thesis: verum

end;

0 in dom J by A1, Th39;
then 0 + n in dom (Relocated (J,n)) by COMPOS_1:118;
then A25: 0 + n in dom (ProgramPart (Relocated (J,n))) by COMPOS_1:16;
V: (ProgramPart s1) /. (IC s1) = s1 . (IC s1) by COMPOS_1:38;
U: (ProgramPart s2) /. (IC s2) = s2 . (IC s2) by COMPOS_1:38;
u: Comput ((ProgramPart s1),s1,0) = s1 by EXTPRO_1:3;
v: Comput ((ProgramPart s2),s2,0) = s2 by EXTPRO_1:3;
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 A7, A9, V, COMPOS_1:122
.= (ProgramPart (Relocated (J,n))) . n by FUNCT_1:72
.= CurInstr ((ProgramPart s2),s2) by A10, A11, A25, U, GRFUNC_1:8
.= CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,0))),(Comput ((ProgramPart s2),s2,0))) by v ;
then A26: S₁[ 0 ] by A11, A8, A13, EXTPRO_1:3;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A26, A14);
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: verum