let s be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA
for a being Int-Location st not I destroys a & I is_closed_on s holds
for k being Element of NAT holds (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)) . a = s . a
let I be Program of SCM+FSA; :: thesis: for a being Int-Location st not I destroys a & I is_closed_on s holds
for k being Element of NAT holds (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)) . a = s . a
let a be Int-Location ; :: thesis: ( not I destroys a & I is_closed_on s implies for k being Element of NAT holds (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)) . a = s . a )
assume A1: not I destroys a ; :: thesis: ( not I is_closed_on s or for k being Element of NAT holds (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)) . a = s . a )
defpred S₁[ Nat] means (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),$1)) . a = s . a;
dom I misses dom (Start-At (0,SCM+FSA)) by COMPOS_1:140;
then ( I +* (Start-At (0,SCM+FSA)) c= s +* (I +* (Start-At (0,SCM+FSA))) & I c= I +* (Start-At (0,SCM+FSA)) ) by FUNCT_4:26, FUNCT_4:33;
then A2: I c= s +* (I +* (Start-At (0,SCM+FSA))) by XBOOLE_1:1;
assume A3: I is_closed_on s ; :: thesis: for k being Element of NAT holds (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)) . a = s . a

A4: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A5: S₁[k] ; :: thesis: S₁[k + 1]
set l = IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k));
A6: IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)) in dom I by A3, Def7;
then (s +* (I +* (Start-At (0,SCM+FSA)))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))) = I . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))) by A2, GRFUNC_1:8;
then (s +* (I +* (Start-At (0,SCM+FSA)))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))) in rng I by A6, FUNCT_1:def 5;
then A7: not (s +* (I +* (Start-At (0,SCM+FSA)))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))) destroys a by A1, Def4;
(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(k + 1))) . a = (Following ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)))) . a by EXTPRO_1:4
.= (Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)))) . a by AMI_1:123
.= (Exec (((Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)))) . a by COMPOS_1:38
.= (Exec (((s +* (I +* (Start-At (0,SCM+FSA)))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),k)))) . a by AMI_1:54
.= s . a by A5, A7, Th26 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A8: not a in dom (I +* (Start-At (0,SCM+FSA))) by SCMFSA6B:12;
(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),0)) . a = (s +* (I +* (Start-At (0,SCM+FSA)))) . a by EXTPRO_1:3
.= s . a by A8, FUNCT_4:12 ;
then A9: S₁[ 0 ] ;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A9, A4); :: thesis: verum