let s be State of SCM+FSA; :: thesis: for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for k being Nat
for P being Instruction-Sequence of SCM+FSA holds (StepWhile>0 (a,P,s,I)) . (k + 1) = (StepWhile>0 (a,P,((StepWhile>0 (a,P,s,I)) . k),I)) . 1

let I be MacroInstruction of SCM+FSA ; :: thesis: for a being read-write Int-Location
for k being Nat
for P being Instruction-Sequence of SCM+FSA holds (StepWhile>0 (a,P,s,I)) . (k + 1) = (StepWhile>0 (a,P,((StepWhile>0 (a,P,s,I)) . k),I)) . 1

let a be read-write Int-Location; :: thesis: for k being Nat
for P being Instruction-Sequence of SCM+FSA holds (StepWhile>0 (a,P,s,I)) . (k + 1) = (StepWhile>0 (a,P,((StepWhile>0 (a,P,s,I)) . k),I)) . 1

let k be Nat; :: thesis: for P being Instruction-Sequence of SCM+FSA holds (StepWhile>0 (a,P,s,I)) . (k + 1) = (StepWhile>0 (a,P,((StepWhile>0 (a,P,s,I)) . k),I)) . 1
let P be Instruction-Sequence of SCM+FSA; :: thesis: (StepWhile>0 (a,P,s,I)) . (k + 1) = (StepWhile>0 (a,P,((StepWhile>0 (a,P,s,I)) . k),I)) . 1
set sk = (StepWhile>0 (a,P,s,I)) . k;
set sk0 = (StepWhile>0 (a,P,((StepWhile>0 (a,P,s,I)) . k),I)) . 0;
(StepWhile>0 (a,P,((StepWhile>0 (a,P,s,I)) . k),I)) . 0 = (StepWhile>0 (a,P,s,I)) . k by Def1;
hence (StepWhile>0 (a,P,s,I)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(Initialized ((StepWhile>0 (a,P,((StepWhile>0 (a,P,s,I)) . k),I)) . 0)),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialized ((StepWhile>0 (a,P,((StepWhile>0 (a,P,s,I)) . k),I)) . 0)))) + 2)) by Def1
.= (StepWhile>0 (a,P,((StepWhile>0 (a,P,s,I)) . k),I)) . (0 + 1) by Def1
.= (StepWhile>0 (a,P,((StepWhile>0 (a,P,s,I)) . k),I)) . 1 ;
:: thesis: verum