let s1, s2 be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA st I +* (Start-At (0,SCM+FSA)) c= s1 & I is_closed_on 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 s1),(Comput ((ProgramPart s1),s1,i)))),n) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) & DataPart (Comput ((ProgramPart s1),s1,i)) = DataPart (Comput ((ProgramPart s2),s2,i)) )
let I be Program of SCM+FSA; :: thesis: ( I +* (Start-At (0,SCM+FSA)) c= s1 & I is_closed_on s1 implies 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 s1),(Comput ((ProgramPart s1),s1,i)))),n) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) & DataPart (Comput ((ProgramPart s1),s1,i)) = DataPart (Comput ((ProgramPart s2),s2,i)) ) )
assume A1: I +* (Start-At (0,SCM+FSA)) c= s1 ; :: thesis: ( not I is_closed_on s1 or 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 s1),(Comput ((ProgramPart s1),s1,i)))),n) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) & DataPart (Comput ((ProgramPart s1),s1,i)) = DataPart (Comput ((ProgramPart s2),s2,i)) ) )
assume A2: I is_closed_on s1 ; :: thesis: 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 s1),(Comput ((ProgramPart s1),s1,i)))),n) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) & DataPart (Comput ((ProgramPart s1),s1,i)) = DataPart (Comput ((ProgramPart s2),s2,i)) )
let n be Element of NAT ; :: thesis: ( ProgramPart (Relocated (I,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 s1),(Comput ((ProgramPart s1),s1,i)))),n) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) & DataPart (Comput ((ProgramPart s1),s1,i)) = DataPart (Comput ((ProgramPart s2),s2,i)) ) )
A3: IC SCM+FSA in dom (I +* (Start-At (0,SCM+FSA))) by COMPOS_1:141;
A4: I c= I +* (Start-At (0,SCM+FSA)) by SCMFSA8A:9;
then A5: dom I c= dom (I +* (Start-At (0,SCM+FSA))) by GRFUNC_1:8;
defpred S₁[ Nat] means ( (IC (Comput ((ProgramPart s1),s1,$1))) + n = IC (Comput ((ProgramPart s2),s2,$1)) & IncAddr ((CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,$1)))),n) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,$1))) & DataPart (Comput ((ProgramPart s1),s1,$1)) = DataPart (Comput ((ProgramPart s2),s2,$1)) );
assume A6: ProgramPart (Relocated (I,n)) c= s2 ; :: thesis: ( not IC s2 = n or not DataPart s1 = DataPart s2 or for i being Element of NAT holds
( (IC (Comput ((ProgramPart s1),s1,i))) + n = IC (Comput ((ProgramPart s2),s2,i)) & IncAddr ((CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i)))),n) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) & DataPart (Comput ((ProgramPart s1),s1,i)) = DataPart (Comput ((ProgramPart s2),s2,i)) ) )
A7: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
T: ProgramPart s2 = ProgramPart (Comput ((ProgramPart s2),s2,k)) by AMI_1:123;
A8: Comput ((ProgramPart s1),s1,(k + 1)) = Following ((ProgramPart s1),(Comput ((ProgramPart s1),s1,k))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,k)))),(Comput ((ProgramPart s1),s1,k))) ;
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 ;
A9: 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 +* (I +* (Start-At (0,SCM+FSA))) = s1 by A1, FUNCT_4:79;
then A11: IC (Comput ((ProgramPart s1),s1,(k + 1))) in dom I by A2, SCMFSA7B:def 7;
assume A12: S₁[k] ; :: thesis: S₁[k + 1]
hence A13: (IC (Comput ((ProgramPart s1),s1,(k + 1)))) + n = IC (Comput ((ProgramPart s2),s2,(k + 1))) by A8, A9, T, SCMFSA6A:41; :: thesis: ( IncAddr ((CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,(k + 1))))),n) = CurInstr ((ProgramPart s2),(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 (I,n)) by A11, COMPOS_1:118;
then IC (Comput ((ProgramPart s2),s2,(k + 1))) in (dom (Relocated (I,n))) /\ NAT by XBOOLE_0:def 4;
then A14: IC (Comput ((ProgramPart s2),s2,(k + 1))) in dom (ProgramPart (Relocated (I,n))) by RELAT_1:90;
dom (ProgramPart I) = (dom I) /\ NAT by RELAT_1:90;
then A15: l in dom (ProgramPart I) by A11, XBOOLE_0:def 4;
A16: I c= I +* (Start-At (0,SCM+FSA)) by SCMFSA8A:9;
then A17: dom I c= dom (I +* (Start-At (0,SCM+FSA))) by GRFUNC_1:8;
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;
TX1: ProgramPart s1 = ProgramPart (Comput ((ProgramPart s1),s1,(k + 1))) by AMI_1:123;
TX2: ProgramPart s2 = ProgramPart (Comput ((ProgramPart s2),s2,(k + 1))) by AMI_1:123;
j = s1 . (IC (Comput ((ProgramPart s1),s1,(k + 1)))) by Y, AMI_1:54
.= (I +* (Start-At (0,SCM+FSA))) . (IC (Comput ((ProgramPart s1),s1,(k + 1)))) by A1, A17, A11, GRFUNC_1:8
.= I . l by A16, A11, GRFUNC_1:8 ;
hence IncAddr ((CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,(k + 1))))),n) = (Relocated (I,n)) . (l + n) by A15, TX1, COMPOS_1:122
.= (ProgramPart (Relocated (I,n))) . (IC (Comput ((ProgramPart s2),s2,(k + 1)))) by A13, FUNCT_1:72
.= s2 . (IC (Comput ((ProgramPart s2),s2,(k + 1)))) by A6, A14, GRFUNC_1:8
.= CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,(k + 1)))) by Z, TX2, 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 A12, A8, A9, T, SCMFSA6A:41; :: thesis: verum

end;

A18: IC (Comput ((ProgramPart s1),s1,0)) = s1 . (IC SCM+FSA) by EXTPRO_1:3
.= IC (I +* (Start-At (0,SCM+FSA))) by A1, A3, GRFUNC_1:8
.= 0 by COMPOS_1:142 ;
assume A19: IC s2 = n ; :: thesis: ( not DataPart s1 = DataPart s2 or for i being Element of NAT holds
( (IC (Comput ((ProgramPart s1),s1,i))) + n = IC (Comput ((ProgramPart s2),s2,i)) & IncAddr ((CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i)))),n) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) & DataPart (Comput ((ProgramPart s1),s1,i)) = DataPart (Comput ((ProgramPart s2),s2,i)) ) )
A20: 0 in dom I by A2, Th3;
then 0 + n in dom (Relocated (I,n)) by COMPOS_1:118;
then A21: 0 + n in dom (ProgramPart (Relocated (I,n))) by COMPOS_1:16;
IC SCM+FSA in dom (I +* (Start-At (0,SCM+FSA))) by COMPOS_1:141;
then A22: s1 . (IC s1) = s1 . (IC (I +* (Start-At (0,SCM+FSA)))) by A1, GRFUNC_1:8
.= s1 . 0 by COMPOS_1:142
.= (I +* (Start-At (0,SCM+FSA))) . 0 by A1, A5, A20, GRFUNC_1:8
.= I . 0 by A4, A20, GRFUNC_1:8 ;
ProgramPart I = I by RELAT_1:209;
then A23: 0 in dom (ProgramPart I) by A2, Th3;
assume 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 s1),(Comput ((ProgramPart s1),s1,i)))),n) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) & DataPart (Comput ((ProgramPart s1),s1,i)) = DataPart (Comput ((ProgramPart s2),s2,i)) )
then A24: DataPart (Comput ((ProgramPart s1),s1,0)) = DataPart s2 by EXTPRO_1:3
.= DataPart (Comput ((ProgramPart s2),s2,0)) by EXTPRO_1:3 ;
let i be Element of NAT ; :: thesis: ( (IC (Comput ((ProgramPart s1),s1,i))) + n = IC (Comput ((ProgramPart s2),s2,i)) & IncAddr ((CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i)))),n) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) & DataPart (Comput ((ProgramPart s1),s1,i)) = DataPart (Comput ((ProgramPart s2),s2,i)) )
Y: (ProgramPart s1) /. (IC s1) = s1 . (IC s1) by COMPOS_1:38;
V: (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 s1),(Comput ((ProgramPart s1),s1,0)))),n) = IncAddr ((CurInstr ((ProgramPart s1),s1)),n) by u
.= (Relocated (I,n)) . (0 + n) by A22, A23, Y, COMPOS_1:122
.= (ProgramPart (Relocated (I,n))) . n by FUNCT_1:72
.= CurInstr ((ProgramPart s2),s2) by A6, A19, A21, V, GRFUNC_1:8
.= CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,0))) by v ;
then A25: S₁[ 0 ] by A19, A18, A24, EXTPRO_1:3;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A25, A7);
hence ( (IC (Comput ((ProgramPart s1),s1,i))) + n = IC (Comput ((ProgramPart s2),s2,i)) & IncAddr ((CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i)))),n) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) & DataPart (Comput ((ProgramPart s1),s1,i)) = DataPart (Comput ((ProgramPart s2),s2,i)) ) ; :: thesis: verum