let s be State of SCM+FSA; :: thesis:
let p be Instruction-Sequence of SCM+FSA; :: thesis:
let a be read-write Int-Location; :: thesis:
let J be good really-closed MacroInstruction of SCM+FSA ; :: thesis:
set I = J;
set ST = StepTimes (a,J,p,s);
set Is = Initialized s;
defpred S₁[ Nat] means ( $1 <= s . a implies ((StepTimes (a,J,p,s)) . $1) . (intloc 0) = 1 );
assume A1: not J destroys a ; :: thesis:
assume A2: ProperTimesBody a,J,s,p ; :: thesis:
A3: for k being Nat st S₁[k] holds
S₁[k + 1]

let k be Nat; :: thesis:
assume that
A4: ( k <= s . a implies ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 ) and
A5: k + 1 <= s . a ; :: thesis:
reconsider sa = s . a as Element of NAT by A5, INT_1:3;
A6: k < sa by A5, NAT_1:13;
then J is_halting_on (StepTimes (a,J,p,s)) . k,p +* (Times (a,J)) by A2;
hence ((StepTimes (a,J,p,s)) . (k + 1)) . (intloc 0) = 1 by A4, A6, Th48, A1; :: thesis:

end;

assume 0 <= s . a ; :: thesis:
thus ((StepTimes (a,J,p,s)) . 0) . (intloc 0) = (Initialized s) . (intloc 0) by SCMFSA_9:def 5
.= 1 by SCMFSA_M:9 ; :: thesis:

end;

thus for k being Nat holds S₁[k] from NAT_1:sch 2(A7, A3); :: thesis: