let p1, p2 be Instruction-Sequence of SCM+FSA; :: thesis: for s1, s2 being State of SCM+FSA
for a being read-write Int-Location
for I being Program of SCM+FSA st DataPart s1 = DataPart s2 & ProperBodyWhile>0 a,I,s1,p1 holds
ProperBodyWhile>0 a,I,s2,p2
let s1, s2 be State of SCM+FSA; :: thesis: for a being read-write Int-Location
for I being Program of SCM+FSA st DataPart s1 = DataPart s2 & ProperBodyWhile>0 a,I,s1,p1 holds
ProperBodyWhile>0 a,I,s2,p2
let a be read-write Int-Location ; :: thesis: for I being Program of SCM+FSA st DataPart s1 = DataPart s2 & ProperBodyWhile>0 a,I,s1,p1 holds
ProperBodyWhile>0 a,I,s2,p2
let I be Program of SCM+FSA; :: thesis: ( DataPart s1 = DataPart s2 & ProperBodyWhile>0 a,I,s1,p1 implies ProperBodyWhile>0 a,I,s2,p2 )
assume that
A1: DataPart s1 = DataPart s2 and
A2: ProperBodyWhile>0 a,I,s1,p1 ; :: thesis: ProperBodyWhile>0 a,I,s2,p2
let k be Element of NAT ; :: according to SCMFSA9A:def 4 :: thesis: ( ((StepWhile>0 (a,I,p2,s2)) . k) . a > 0 implies ( I is_closed_on (StepWhile>0 (a,I,p2,s2)) . k,p2 +* (while>0 (a,I)) & I is_halting_on (StepWhile>0 (a,I,p2,s2)) . k,p2 +* (while>0 (a,I)) ) )
assume A3: ((StepWhile>0 (a,I,p2,s2)) . k) . a > 0 ; :: thesis: ( I is_closed_on (StepWhile>0 (a,I,p2,s2)) . k,p2 +* (while>0 (a,I)) & I is_halting_on (StepWhile>0 (a,I,p2,s2)) . k,p2 +* (while>0 (a,I)) )
A4: DataPart ((StepWhile>0 (a,I,p2,s2)) . k) = DataPart ((StepWhile>0 (a,I,p1,s1)) . k) by A1, A2, Th40;
then ((StepWhile>0 (a,I,p1,s1)) . k) . a > 0 by A3, SCMFSA6A:7;
then ( I is_closed_on (StepWhile>0 (a,I,p1,s1)) . k,p1 +* (while>0 (a,I)) & I is_halting_on (StepWhile>0 (a,I,p1,s1)) . k,p1 +* (while>0 (a,I)) ) by A2, Def4;
hence ( I is_closed_on (StepWhile>0 (a,I,p2,s2)) . k,p2 +* (while>0 (a,I)) & I is_halting_on (StepWhile>0 (a,I,p2,s2)) . k,p2 +* (while>0 (a,I)) ) by A4, SCMFSA8B:5; :: thesis: verum