now :: thesis: for s being All-State of SumTuring
for t being Tape of SumTuring
for h1 being Element of NAT
for x being FinSequence of NAT st x in dom [+] & s = [ the InitS of SumTuring,h1,t] & t storeData <*h1*> ^ x holds
( s is Accept-Halt & ex h2 being Element of omega ex y being Element of omega st
( () `2_3 = h2 & y = [+] . x & () `3_3 storeData <*h2*> ^ <*y*> ) )
let s be All-State of SumTuring; :: thesis: for t being Tape of SumTuring
for h1 being Element of NAT
for x being FinSequence of NAT st x in dom [+] & s = [ the InitS of SumTuring,h1,t] & t storeData <*h1*> ^ x holds
( s is Accept-Halt & ex h2 being Element of omega ex y being Element of omega st
( () `2_3 = h2 & y = [+] . x & () `3_3 storeData <*h2*> ^ <*y*> ) )

let t be Tape of SumTuring; :: thesis: for h1 being Element of NAT
for x being FinSequence of NAT st x in dom [+] & s = [ the InitS of SumTuring,h1,t] & t storeData <*h1*> ^ x holds
( s is Accept-Halt & ex h2 being Element of omega ex y being Element of omega st
( () `2_3 = h2 & y = [+] . x & () `3_3 storeData <*h2*> ^ <*y*> ) )

let h1 be Element of NAT ; :: thesis: for x being FinSequence of NAT st x in dom [+] & s = [ the InitS of SumTuring,h1,t] & t storeData <*h1*> ^ x holds
( s is Accept-Halt & ex h2 being Element of omega 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 SumTuring,h1,t] & t storeData <*h1*> ^ x implies ( s is Accept-Halt & ex h2 being Element of omega ex y being Element of omega st
( () `2_3 = h2 & y = [+] . x & () `3_3 storeData <*h2*> ^ <*y*> ) ) )

assume that
A1: x in dom [+] and
A2: s = [ the InitS of SumTuring,h1,t] and
A3: t storeData <*h1*> ^ x ; :: thesis: ( s is Accept-Halt & ex h2 being Element of omega ex y being Element of omega st
( () `2_3 = h2 & y = [+] . x & () `3_3 storeData <*h2*> ^ <*y*> ) )

x is Tuple of 2, NAT by ;
then consider i, j being Element of NAT such that
A4: x = <*i,j*> by FINSEQ_2:100;
A5: s = [0,h1,t] by ;
A6: <*h1*> ^ x = <*h1,i,j*> by ;
hence s is Accept-Halt by A3, A5, Th27; :: thesis: ex h2 being Element of omega ex y being Element of omega st
( () `2_3 = h2 & y = [+] . x & () `3_3 storeData <*h2*> ^ <*y*> )

take h2 = 1 + h1; :: thesis: ex y being Element of omega st
( () `2_3 = h2 & y = [+] . x & () `3_3 storeData <*h2*> ^ <*y*> )

take y = i + j; :: thesis: ( () `2_3 = h2 & y = [+] . x & () `3_3 storeData <*h2*> ^ <*y*> )
t storeData <*h1,i,j*> by ;
hence (Result s) `2_3 = h2 by ; :: thesis: ( y = [+] . x & () `3_3 storeData <*h2*> ^ <*y*> )
thus y = [+] . x by ; :: thesis: () `3_3 storeData <*h2*> ^ <*y*>
(Result s) `3_3 storeData <*(1 + h1),(i + j)*> by A3, A5, A6, Th27;
hence (Result s) `3_3 storeData <*h2*> ^ <*y*> ; :: thesis: verum
end;
hence SumTuring computes [+] ; :: thesis: verum