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

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

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

let I be MacroInstruction of SCM+FSA ; :: thesis: for a being read-write Int-Location holds (StepWhile>0 (a,I,P,s)) . (k + 1) = (StepWhile>0 (a,I,P,((StepWhile>0 (a,I,P,s)) . k))) . 1
let a be read-write Int-Location; :: thesis: (StepWhile>0 (a,I,P,s)) . (k + 1) = (StepWhile>0 (a,I,P,((StepWhile>0 (a,I,P,s)) . k))) . 1
set sk = (StepWhile>0 (a,I,P,s)) . k;
set sk0 = (StepWhile>0 (a,I,P,((StepWhile>0 (a,I,P,s)) . k))) . 0;
(StepWhile>0 (a,I,P,((StepWhile>0 (a,I,P,s)) . k))) . 0 = (StepWhile>0 (a,I,P,s)) . k by Def2;
hence (StepWhile>0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(Initialize ((StepWhile>0 (a,I,P,((StepWhile>0 (a,I,P,s)) . k))) . 0)),((LifeSpan (((P +* (while>0 (a,I))) +* I),(Initialize ((StepWhile>0 (a,I,P,((StepWhile>0 (a,I,P,s)) . k))) . 0)))) + 2)) by Def2
.= (StepWhile>0 (a,I,P,((StepWhile>0 (a,I,P,s)) . k))) . (0 + 1) by Def2
.= (StepWhile>0 (a,I,P,((StepWhile>0 (a,I,P,s)) . k))) . 1 ;
:: thesis: verum