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_pseudo-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 st i < pseudo-LifeSpan s1,I holds
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) ) & ( for i being Element of NAT st i <= pseudo-LifeSpan s1,I holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (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_pseudo-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 st i < pseudo-LifeSpan s1,I holds
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) ) & ( for i being Element of NAT st i <= pseudo-LifeSpan s1,I holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (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_pseudo-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 st i < pseudo-LifeSpan s1,I holds
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) ) & ( for i being Element of NAT st i <= pseudo-LifeSpan s1,I holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) ) ) ) )
assume A2: I is_pseudo-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 st i < pseudo-LifeSpan s1,I holds
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) ) & ( for i being Element of NAT st i <= pseudo-LifeSpan s1,I holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (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 st i < pseudo-LifeSpan s1,I holds
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) ) & ( for i being Element of NAT st i <= pseudo-LifeSpan s1,I holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) ) ) ) )
assume A3: ProgramPart (Relocated I,n) c= s2 ; :: thesis: ( not IC s2 = n or not DataPart s1 = DataPart s2 or ( ( for i being Element of NAT st i < pseudo-LifeSpan s1,I holds
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) ) & ( for i being Element of NAT st i <= pseudo-LifeSpan s1,I holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) ) ) ) )
defpred S1[ Nat] means ( $1 <= pseudo-LifeSpan s1,I implies ( (IC (Comput (ProgramPart s1),s1,$1)) + n = IC (Comput (ProgramPart s2),s2,$1) & DataPart (Comput (ProgramPart s1),s1,$1) = DataPart (Comput (ProgramPart s2),s2,$1) ) );
assume A4: IC s2 = n ; :: thesis: ( not DataPart s1 = DataPart s2 or ( ( for i being Element of NAT st i < pseudo-LifeSpan s1,I holds
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) ) & ( for i being Element of NAT st i <= pseudo-LifeSpan s1,I holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) ) ) ) )
assume A5: DataPart s1 = DataPart s2 ; :: thesis: ( ( for i being Element of NAT st i < pseudo-LifeSpan s1,I holds
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) ) & ( for i being Element of NAT st i <= pseudo-LifeSpan s1,I holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) ) ) )
hereby :: thesis: for i being Element of NAT st i <= pseudo-LifeSpan s1,I holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) )
defpred S2[ Nat] means ( $1 < pseudo-LifeSpan s1,I implies ( (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) ) );
let i be Element of NAT ; :: thesis: ( i < pseudo-LifeSpan s1,I implies 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) )
assume A7: i < pseudo-LifeSpan s1,I ; :: thesis: 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)
A8: for k being Element of NAT st S2[k] holds
S2[k + 1]
proof
A9: I c= I +* (Start-At 0 ,SCM+FSA ) by SCMFSA8A:9;
then A10: dom I c= dom (I +* (Start-At 0 ,SCM+FSA )) by GRFUNC_1:8;
let k be Element of NAT ; :: thesis: ( S2[k] implies S2[k + 1] )
assume A11: S2[k] ; :: thesis: S2[k + 1]
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;
assume A12: k + 1 < pseudo-LifeSpan s1,I ; :: thesis: ( (IC (Comput (ProgramPart s1),s1,(k + 1))) + n = IC (Comput (ProgramPart s2),s2,(k + 1)) & 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)) )
T: ProgramPart s1 = ProgramPart (Comput (ProgramPart s1),s1,k) by AMI_1:123;
A14: 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 ;
s1 +* (I +* (Start-At 0 ,SCM+FSA )) = s1 by A1, FUNCT_4:79;
then A15: IC (Comput (ProgramPart s1),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;
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: k + 0 < k + 1 by XREAL_1:8;
hence A19: (IC (Comput (ProgramPart s1),s1,(k + 1))) + n = IC (Comput (ProgramPart s2),s2,(k + 1)) by A11, A12, A14, A17, SCMFSA6A:41, XXREAL_0:2; :: 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 I,n) by A15, AMISTD_2:71;
then IC (Comput (ProgramPart s2),s2,(k + 1)) in (dom (Relocated I,n)) /\ NAT by XBOOLE_0:def 4;
then A20: IC (Comput (ProgramPart s2),s2,(k + 1)) in dom (ProgramPart (Relocated I,n)) by RELAT_1:90;
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
.= (I +* (Start-At 0 ,SCM+FSA )) . (IC (Comput (ProgramPart s1),s1,(k + 1))) by A1, A10, A15, GRFUNC_1:8
.= I . l by A9, A15, GRFUNC_1:8 ;
hence IncAddr (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,(k + 1))),(Comput (ProgramPart s1),s1,(k + 1))),n = (Relocated I,n) . (l + n) by A16, AMISTD_2:76
.= (ProgramPart (Relocated I,n)) . (IC (Comput (ProgramPart s2),s2,(k + 1))) by A19, FUNCT_1:72
.= s2 . (IC (Comput (ProgramPart s2),s2,(k + 1))) by A3, A20, GRFUNC_1:8
.= CurInstr (ProgramPart (Comput (ProgramPart s2),s2,(k + 1))),(Comput (ProgramPart s2),s2,(k + 1)) by Z, 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 A11, A12, A18, A14, A17, SCMFSA6A:41, XXREAL_0:2; :: thesis: verum
end;
A21: S2[ 0 ]
proof
A22: IC (Comput (ProgramPart (s1 +* (I +* (Start-At 0 ,SCM+FSA )))),(s1 +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) = IC (s1 +* (I +* (Start-At 0 ,SCM+FSA ))) by AMI_1:13
.= IC ((s1 +* I) +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:15
.= 0 by FUNCT_4:121 ;
assume 0 < pseudo-LifeSpan s1,I ; :: thesis: ( (IC (Comput (ProgramPart s1),s1,0 )) + n = IC (Comput (ProgramPart s2),s2,0 ) & IncAddr (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,0 )),(Comput (ProgramPart s1),s1,0 )),n = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,0 )),(Comput (ProgramPart s2),s2,0 ) & DataPart (Comput (ProgramPart s1),s1,0 ) = DataPart (Comput (ProgramPart s2),s2,0 ) )
then A23: 0 in dom I by A2, A22, SCMFSA8A:def 5;
then A24: 0 in dom (ProgramPart I) by RELAT_1:209;
A25: IC SCM+FSA in dom (I +* (Start-At 0 ,SCM+FSA )) by SF_MASTR:65;
u: Comput (ProgramPart s1),s1,0 = s1 by AMI_1:13;
v: Comput (ProgramPart s2),s2,0 = s2 by AMI_1:13;
IC (Comput (ProgramPart s1),s1,0 ) = s1 . (IC SCM+FSA ) by AMI_1:13
.= (I +* (Start-At 0 ,SCM+FSA )) . (IC SCM+FSA ) by A1, A25, GRFUNC_1:8
.= 0 by SF_MASTR:66 ;
hence (IC (Comput (ProgramPart s1),s1,0 )) + n = IC (Comput (ProgramPart s2),s2,0 ) by A4, AMI_1:13; :: thesis: ( IncAddr (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,0 )),(Comput (ProgramPart s1),s1,0 )),n = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,0 )),(Comput (ProgramPart s2),s2,0 ) & DataPart (Comput (ProgramPart s1),s1,0 ) = DataPart (Comput (ProgramPart s2),s2,0 ) )
A26: I c= I +* (Start-At 0 ,SCM+FSA ) by SCMFSA8A:9;
then A27: dom I c= dom (I +* (Start-At 0 ,SCM+FSA )) by GRFUNC_1:8;
0 + n in dom (Relocated I,n) by A23, AMISTD_2:71;
then A28: 0 + n in dom (ProgramPart (Relocated I,n)) by COMPOS_1:16;
IC SCM+FSA in dom (I +* (Start-At 0 ,SCM+FSA )) by SF_MASTR:65;
then A29: s1 . (IC s1) = s1 . ((I +* (Start-At 0 ,SCM+FSA )) . (IC SCM+FSA )) by A1, GRFUNC_1:8
.= s1 . 0 by SF_MASTR:66
.= (I +* (Start-At 0 ,SCM+FSA )) . 0 by A1, A27, A23, GRFUNC_1:8
.= I . 0 by A26, A23, GRFUNC_1:8 ;
Y: (ProgramPart s1) /. (IC s1) = s1 . (IC s1) by COMPOS_1:38;
Z: (ProgramPart s2) /. (IC s2) = s2 . (IC s2) by COMPOS_1:38;
thus IncAddr (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,0 )),(Comput (ProgramPart s1),s1,0 )),n = IncAddr (CurInstr (ProgramPart s1),s1),n by u
.= (Relocated I,n) . (0 + n) by A29, A24, Y, AMISTD_2:76
.= (ProgramPart (Relocated I,n)) . n by FUNCT_1:72
.= CurInstr (ProgramPart s2),s2 by A3, A4, A28, Z, GRFUNC_1:8
.= CurInstr (ProgramPart (Comput (ProgramPart s2),s2,0 )),(Comput (ProgramPart s2),s2,0 ) by v ; :: thesis: DataPart (Comput (ProgramPart s1),s1,0 ) = DataPart (Comput (ProgramPart s2),s2,0 )
thus DataPart (Comput (ProgramPart s1),s1,0 ) = DataPart s2 by A5, AMI_1:13
.= DataPart (Comput (ProgramPart s2),s2,0 ) by AMI_1:13 ; :: thesis: verum
end;
for k being Element of NAT holds S2[k] from NAT_1:sch 1(A21, A8);
hence 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) by A7; :: thesis: verum
end;
A30: 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] )
assume A31: S1[k] ; :: thesis: S1[k + 1]
set i = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k);
T: ProgramPart s2 = ProgramPart (Comput (ProgramPart s2),s2,k) by AMI_1:123;
A32: 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 ;
assume A33: k + 1 <= pseudo-LifeSpan s1,I ; :: thesis: ( (IC (Comput (ProgramPart s1),s1,(k + 1))) + n = IC (Comput (ProgramPart s2),s2,(k + 1)) & DataPart (Comput (ProgramPart s1),s1,(k + 1)) = DataPart (Comput (ProgramPart s2),s2,(k + 1)) )
then A34: k + 1 <= (pseudo-LifeSpan s1,I) + 1 by NAT_1:12;
A35: k < pseudo-LifeSpan s1,I by A33, NAT_1:13;
T: ProgramPart s1 = ProgramPart (Comput (ProgramPart s1),s1,k) by AMI_1:123;
A36: 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 ;
hence (IC (Comput (ProgramPart s1),s1,(k + 1))) + n = IC (Exec (IncAddr (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k)),n),(Comput (ProgramPart s2),s2,k)) by A31, A34, SCMFSA6A:41, XREAL_1:8
.= IC (Comput (ProgramPart s2),s2,(k + 1)) by A6, A35, A32 ;
:: thesis: DataPart (Comput (ProgramPart s1),s1,(k + 1)) = DataPart (Comput (ProgramPart s2),s2,(k + 1))
thus DataPart (Comput (ProgramPart s1),s1,(k + 1)) = DataPart (Exec (IncAddr (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k)),n),(Comput (ProgramPart s2),s2,k)) by A31, A34, A36, SCMFSA6A:41, XREAL_1:8
.= DataPart (Comput (ProgramPart s2),s2,(k + 1)) by A6, A35, A32 ; :: thesis: verum
end;
let i be Element of NAT ; :: thesis: ( i <= pseudo-LifeSpan s1,I implies ( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) ) )
assume A37: i <= pseudo-LifeSpan s1,I ; :: thesis: ( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) )
A38: S1[ 0 ]
proof
assume 0 <= pseudo-LifeSpan s1,I ; :: thesis: ( (IC (Comput (ProgramPart s1),s1,0 )) + n = IC (Comput (ProgramPart s2),s2,0 ) & DataPart (Comput (ProgramPart s1),s1,0 ) = DataPart (Comput (ProgramPart s2),s2,0 ) )
A39: IC SCM+FSA in dom (I +* (Start-At 0 ,SCM+FSA )) by SF_MASTR:65;
IC (Comput (ProgramPart s1),s1,0 ) = s1 . (IC SCM+FSA ) by AMI_1:13
.= (I +* (Start-At 0 ,SCM+FSA )) . (IC SCM+FSA ) by A1, A39, GRFUNC_1:8
.= 0 by SF_MASTR:66 ;
hence (IC (Comput (ProgramPart s1),s1,0 )) + n = IC (Comput (ProgramPart s2),s2,0 ) by A4, AMI_1:13; :: thesis: DataPart (Comput (ProgramPart s1),s1,0 ) = DataPart (Comput (ProgramPart s2),s2,0 )
thus DataPart (Comput (ProgramPart s1),s1,0 ) = DataPart s2 by A5, AMI_1:13
.= DataPart (Comput (ProgramPart s2),s2,0 ) by AMI_1:13 ; :: thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch 1(A38, A30);
 hence ( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) ) by A37; :: thesis: verum