let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: for s1 being 0 -started State of SCM+FSA
for s2 being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s1,P1 & I c= P1 holds
for n being Element of NAT st IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
let s1 be 0 -started State of SCM+FSA; :: thesis: for s2 being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s1,P1 & I c= P1 holds
for n being Element of NAT st IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
let s2 be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA st I is_closed_on s1,P1 & I c= P1 holds
for n being Element of NAT st IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
let I be Program of SCM+FSA; :: thesis: ( I is_closed_on s1,P1 & I c= P1 implies for n being Element of NAT st IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) )
A1: Start-At (0,SCM+FSA) c= s1 by MEMSTR_0:29;
assume A2: I is_closed_on s1,P1 ; :: thesis: ( not I c= P1 or for n being Element of NAT st IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) )
assume A3: I c= P1 ; :: thesis: for n being Element of NAT st IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
let n be Element of NAT ; :: thesis: ( IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 implies for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) )
A4: IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
defpred S₁[ Nat] means ( (IC (Comput (P1,s1,$1))) + n = IC (Comput (P2,s2,$1)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,$1)))),n) = CurInstr (P2,(Comput (P2,s2,$1))) & DataPart (Comput (P1,s1,$1)) = DataPart (Comput (P2,s2,$1)) );
A5: IC (Comput (P1,s1,0)) = IC s1
.= IC (Start-At (0,SCM+FSA)) by A1, A4, GRFUNC_1:2
.= 0 by FUNCOP_1:72 ;
assume A6: IC s2 = n ; :: thesis: ( not DataPart s1 = DataPart s2 or not Reloc (I,n) c= P2 or for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) )
A7: 0 in dom I by A2, Th1;
then A8: 0 + n in dom (Reloc (I,n)) by COMPOS_1:46;
IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
then A9: P1 . (IC s1) = P1 . (IC (Start-At (0,SCM+FSA))) by A1, GRFUNC_1:2
.= P1 . 0 by FUNCOP_1:72
.= I . 0 by A7, A3, GRFUNC_1:2 ;
assume DataPart s1 = DataPart s2 ; :: thesis: ( not Reloc (I,n) c= P2 or for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) )
then A10: DataPart (Comput (P1,s1,0)) = DataPart s2
.= DataPart (Comput (P2,s2,0)) ;
assume A11: Reloc (I,n) c= P2 ; :: thesis: for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
let i be Element of NAT ; :: thesis: ( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
A12: P2 /. (IC s2) = P2 . (IC s2) by PBOOLE:143;
A13: CurInstr (P1,s1) = I . 0 by A9, PBOOLE:143;
IncAddr ((CurInstr (P1,(Comput (P1,s1,0)))),n) = IncAddr ((CurInstr (P1,s1)),n)
.= (Reloc (I,n)) . (0 + n) by A13, A7, COMPOS_1:35
.= CurInstr (P2,s2) by A6, A8, A12, A11, GRFUNC_1:2
.= CurInstr (P2,(Comput (P2,s2,0))) ;
then A14: S₁[ 0 ] by A6, A5, A10;
A15: 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] )
A16: Comput (P1,s1,(k + 1)) = Following (P1,(Comput (P1,s1,k))) by EXTPRO_1:3
.= Exec ((CurInstr (P1,(Comput (P1,s1,k)))),(Comput (P1,s1,k))) ;
reconsider l = IC (Comput (P1,s1,(k + 1))) as Element of NAT ;
reconsider j = CurInstr (P1,(Comput (P1,s1,(k + 1)))) as Instruction of SCM+FSA ;
A17: Comput (P2,s2,(k + 1)) = Following (P2,(Comput (P2,s2,k))) by EXTPRO_1:3
.= Exec ((CurInstr (P2,(Comput (P2,s2,k)))),(Comput (P2,s2,k))) ;
A18: Initialize s1 = s1 by A1, FUNCT_4:98;
A19: P1 = P1 +* I by A3, FUNCT_4:98;
then A20: IC (Comput (P1,s1,(k + 1))) in dom I by A2, A18, SCMFSA7B:def 6;
assume A21: S₁[k] ; :: thesis: S₁[k + 1]
hence A22: (IC (Comput (P1,s1,(k + 1)))) + n = IC (Comput (P2,s2,(k + 1))) by A16, A17, SCMFSA6A:8; :: thesis: ( IncAddr ((CurInstr (P1,(Comput (P1,s1,(k + 1))))),n) = CurInstr (P2,(Comput (P2,s2,(k + 1)))) & DataPart (Comput (P1,s1,(k + 1))) = DataPart (Comput (P2,s2,(k + 1))) )
then A23: IC (Comput (P2,s2,(k + 1))) in dom (Reloc (I,n)) by A20, COMPOS_1:46;
A24: l in dom I by A18, A2, A19, SCMFSA7B:def 6;
j = P1 . (IC (Comput (P1,s1,(k + 1)))) by PBOOLE:143
.= I . l by A20, A3, GRFUNC_1:2 ;
hence IncAddr ((CurInstr (P1,(Comput (P1,s1,(k + 1))))),n) = (Reloc (I,n)) . (l + n) by A24, COMPOS_1:35
.= P2 . (IC (Comput (P2,s2,(k + 1)))) by A23, A22, A11, GRFUNC_1:2
.= CurInstr (P2,(Comput (P2,s2,(k + 1)))) by PBOOLE:143 ;
:: thesis: DataPart (Comput (P1,s1,(k + 1))) = DataPart (Comput (P2,s2,(k + 1)))
thus DataPart (Comput (P1,s1,(k + 1))) = DataPart (Comput (P2,s2,(k + 1))) by A21, A16, A17, SCMFSA6A:8; :: thesis: verum

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A14, A15);
hence ( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) ; :: thesis: verum