let s be State of SCM+FSA ; :: thesis: for a being read-write Int-Location
for I being parahalting Program of SCM+FSA st WithVariantWhile=0 a,I,s holds
( while=0 a,I is_halting_on s & while=0 a,I is_closed_on s )
let a be read-write Int-Location ; :: thesis: for I being parahalting Program of SCM+FSA st WithVariantWhile=0 a,I,s holds
( while=0 a,I is_halting_on s & while=0 a,I is_closed_on s )
let I be parahalting Program of SCM+FSA ; :: thesis: ( WithVariantWhile=0 a,I,s implies ( while=0 a,I is_halting_on s & while=0 a,I is_closed_on s ) )
assume A1: WithVariantWhile=0 a,I,s ; :: thesis: ( while=0 a,I is_halting_on s & while=0 a,I is_closed_on s )
ProperBodyWhile=0 a,I,s

proof

let k be Element of NAT ; :: according to SCMFSA9A:def 1 :: thesis: ( ((StepWhile=0 a,I,s) . k) . a = 0 implies ( I is_closed_on (StepWhile=0 a,I,s) . k & I is_halting_on (StepWhile=0 a,I,s) . k ) )
assume ((StepWhile=0 a,I,s) . k) . a = 0 ; :: thesis: ( I is_closed_on (StepWhile=0 a,I,s) . k & I is_halting_on (StepWhile=0 a,I,s) . k )
thus ( I is_closed_on (StepWhile=0 a,I,s) . k & I is_halting_on (StepWhile=0 a,I,s) . k ) by SCMFSA7B:24, SCMFSA7B:25; :: thesis: verum

end;

hence ( while=0 a,I is_halting_on s & while=0 a,I is_closed_on s ) by A1, Th20; :: thesis: verum