let a be Int-Location ; :: thesis: for I being Program of SCM+FSA
for n being Element of NAT
for s being State of SCM+FSA st I +* (Start-At 0 ,SCM+FSA ) c= s & ( for m being Element of NAT st m < n holds
IC (Comput (ProgramPart s),s,m) in dom I ) & not a in UsedIntLoc I holds
(Comput (ProgramPart s),s,n) . a = s . a
let I be Program of SCM+FSA ; :: thesis: for n being Element of NAT
for s being State of SCM+FSA st I +* (Start-At 0 ,SCM+FSA ) c= s & ( for m being Element of NAT st m < n holds
IC (Comput (ProgramPart s),s,m) in dom I ) & not a in UsedIntLoc I holds
(Comput (ProgramPart s),s,n) . a = s . a
let n be Element of NAT ; :: thesis: for s being State of SCM+FSA st I +* (Start-At 0 ,SCM+FSA ) c= s & ( for m being Element of NAT st m < n holds
IC (Comput (ProgramPart s),s,m) in dom I ) & not a in UsedIntLoc I holds
(Comput (ProgramPart s),s,n) . a = s . a
let s be State of SCM+FSA ; :: thesis: ( I +* (Start-At 0 ,SCM+FSA ) c= s & ( for m being Element of NAT st m < n holds
IC (Comput (ProgramPart s),s,m) in dom I ) & not a in UsedIntLoc I implies (Comput (ProgramPart s),s,n) . a = s . a )
assume that
A1: I +* (Start-At 0 ,SCM+FSA ) c= s and
A2: for m being Element of NAT st m < n holds
IC (Comput (ProgramPart s),s,m) in dom I and
A3: not a in UsedIntLoc I ; :: thesis: (Comput (ProgramPart s),s,n) . a = s . a
defpred S₁[ Nat] means ( $1 <= n implies (Comput (ProgramPart s),s,$1) . a = s . a );
A4: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
set sm = Comput (ProgramPart s),s,m;
assume A5: ( m <= n implies (Comput (ProgramPart s),s,m) . a = s . a ) ; :: thesis: S₁[m + 1]
assume A6: m + 1 <= n ; :: thesis: (Comput (ProgramPart s),s,(m + 1)) . a = s . a
then m < n by NAT_1:13;
then A7: IC (Comput (ProgramPart s),s,m) in dom I by A2;
then A8: I . (IC (Comput (ProgramPart s),s,m)) in rng I by FUNCT_1:def 5;
Y: (ProgramPart (Comput (ProgramPart s),s,m)) /. (IC (Comput (ProgramPart s),s,m)) = (Comput (ProgramPart s),s,m) . (IC (Comput (ProgramPart s),s,m)) by COMPOS_1:38;
dom I misses dom (Start-At 0 ,SCM+FSA ) by Th64;
then I c= I +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:33;
then I c= s by A1, XBOOLE_1:1;
then I c= Comput (ProgramPart s),s,m by AMI_1:81;
then I . (IC (Comput (ProgramPart s),s,m)) = (Comput (ProgramPart s),s,m) . (IC (Comput (ProgramPart s),s,m)) by A7, GRFUNC_1:8;
then UsedIntLoc ((Comput (ProgramPart s),s,m) . (IC (Comput (ProgramPart s),s,m))) c= UsedIntLoc I by A8, Th23;
then A9: not a in UsedIntLoc ((Comput (ProgramPart s),s,m) . (IC (Comput (ProgramPart s),s,m))) by A3;
T: ProgramPart s = ProgramPart (Comput (ProgramPart s),s,m) by AMI_1:123;
thus (Comput (ProgramPart s),s,(m + 1)) . a = (Following (ProgramPart s),(Comput (ProgramPart s),s,m)) . a by AMI_1:14
.= s . a by A5, A6, A9, Th68, Y, T, NAT_1:13 ; :: thesis: verum

end;

A10: S₁[ 0 ] by AMI_1:13;
for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A10, A4);
hence (Comput (ProgramPart s),s,n) . a = s . a ; :: thesis: verum