let n be Nat; :: thesis: ( n >= 1 implies ZeroTuring computes n const 0 )
assume A1: n >= 1 ; :: thesis:
now :: thesis: for s being All-State of ZeroTuring
for t being Tape of ZeroTuring
for h being Element of NAT
for x being FinSequence of NAT st x in dom () & s = [ the InitS of ZeroTuring,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of omega st
( () `2_3 = h2 & y = () . x & () `3_3 storeData <*h2*> ^ <*y*> ) )
set cs = n const 0;
let s be All-State of ZeroTuring; :: thesis: for t being Tape of ZeroTuring
for h being Element of NAT
for x being FinSequence of NAT st x in dom () & s = [ the InitS of ZeroTuring,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of omega st
( () `2_3 = h2 & y = () . x & () `3_3 storeData <*h2*> ^ <*y*> ) )

let t be Tape of ZeroTuring; :: thesis: for h being Element of NAT
for x being FinSequence of NAT st x in dom () & s = [ the InitS of ZeroTuring,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of omega st
( () `2_3 = h2 & y = () . x & () `3_3 storeData <*h2*> ^ <*y*> ) )

let h be Element of NAT ; :: thesis: for x being FinSequence of NAT st x in dom () & s = [ the InitS of ZeroTuring,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of omega st
( () `2_3 = h2 & y = () . x & () `3_3 storeData <*h2*> ^ <*y*> ) )

let x be FinSequence of NAT ; :: thesis: ( x in dom () & s = [ the InitS of ZeroTuring,h,t] & t storeData <*h*> ^ x implies ( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of omega st
( () `2_3 = h2 & y = () . x & () `3_3 storeData <*h2*> ^ <*y*> ) ) )

assume that
A2: x in dom () and
A3: s = [ the InitS of ZeroTuring,h,t] and
A4: t storeData <*h*> ^ x ; :: thesis: ( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of omega st
( () `2_3 = h2 & y = () . x & () `3_3 storeData <*h2*> ^ <*y*> ) )

x in n -tuples_on NAT by A2;
then x in { f where f is Element of NAT * : len f = n } by FINSEQ_2:def 4;
then A6: ex f being Element of NAT * st
( x = f & len f = n ) ;
A7: s = [0,h,t] by ;
hence s is Accept-Halt by A1, A4, A6, Th34; :: thesis: ex h2 being Element of NAT ex y being Element of omega st
( () `2_3 = h2 & y = () . x & () `3_3 storeData <*h2*> ^ <*y*> )

take h2 = h; :: thesis: ex y being Element of omega st
( () `2_3 = h2 & y = () . x & () `3_3 storeData <*h2*> ^ <*y*> )

take y = 0 ; :: thesis: ( () `2_3 = h2 & y = () . x & () `3_3 storeData <*h2*> ^ <*y*> )
thus (Result s) `2_3 = h2 by A1, A4, A6, A7, Th34; :: thesis: ( y = () . x & () `3_3 storeData <*h2*> ^ <*y*> )
thus y = () . x by ; :: thesis: () `3_3 storeData <*h2*> ^ <*y*>
(Result s) `3_3 storeData <*h2,0*> by A1, A4, A6, A7, Th34;
hence (Result s) `3_3 storeData <*h2*> ^ <*y*> ; :: thesis: verum
end;
hence ZeroTuring computes n const 0 ; :: thesis: verum