let n be Element of NAT ; :: thesis: ( n >= 3 implies U3(n)Turing computes n proj 3 )
assume A1: n >= 3 ; :: thesis: U3(n)Turing computes n proj 3

now

let s be All-State of U3(n)Turing ; :: thesis: for t being Tape of U3(n)Turing
for h being Element of NAT
for x being FinSequence of NAT st x in dom (n proj 3) & s = [the InitS of U3(n)Turing ,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2 = h2 & y = (n proj 3) . x & (Result s) `3 storeData <*h2*> ^ <*y*> ) )
let t be Tape of U3(n)Turing ; :: thesis: for h being Element of NAT
for x being FinSequence of NAT st x in dom (n proj 3) & s = [the InitS of U3(n)Turing ,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2 = h2 & y = (n proj 3) . x & (Result s) `3 storeData <*h2*> ^ <*y*> ) )
let h be Element of NAT ; :: thesis: for x being FinSequence of NAT st x in dom (n proj 3) & s = [the InitS of U3(n)Turing ,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2 = h2 & y = (n proj 3) . x & (Result s) `3 storeData <*h2*> ^ <*y*> ) )
let x be FinSequence of NAT ; :: thesis: ( x in dom (n proj 3) & s = [the InitS of U3(n)Turing ,h,t] & t storeData <*h*> ^ x implies ( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2 = h2 & y = (n proj 3) . x & (Result s) `3 storeData <*h2*> ^ <*y*> ) ) )
set pj = n proj 3;
assume A2: ( x in dom (n proj 3) & s = [the InitS of U3(n)Turing ,h,t] & t storeData <*h*> ^ x ) ; :: thesis: ( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2 = h2 & y = (n proj 3) . x & (Result s) `3 storeData <*h2*> ^ <*y*> ) )
arity (n proj 3) = n by COMPUT_1:41;
then A3: dom (n proj 3) c= n -tuples_on NAT by COMPUT_1:24;
then 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 consider f being Element of NAT * such that
A4: ( x = f & len f = n ) ;
A5: s = [0 ,h,t] by A2, Def22;
hence s is Accept-Halt by A1, A2, A4, Th42; :: thesis: ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2 = h2 & y = (n proj 3) . x & (Result s) `3 storeData <*h2*> ^ <*y*> )
take h2 = ((h + (x /. 1)) + (x /. 2)) + 4; :: thesis: ex y being Element of NAT st
( (Result s) `2 = h2 & y = (n proj 3) . x & (Result s) `3 storeData <*h2*> ^ <*y*> )
take y = x /. 3; :: thesis: ( (Result s) `2 = h2 & y = (n proj 3) . x & (Result s) `3 storeData <*h2*> ^ <*y*> )
thus (Result s) `2 = h2 by A1, A2, A4, A5, Th42; :: thesis: ( y = (n proj 3) . x & (Result s) `3 storeData <*h2*> ^ <*y*> )
thus y = x . 3 by A1, A4, FINSEQ_4:24
.= (n proj 3) . x by A2, A3, COMPUT_1:42 ; :: thesis: (Result s) `3 storeData <*h2*> ^ <*y*>
(Result s) `3 storeData <*h2,y*> by A1, A2, A4, A5, Th42;
hence (Result s) `3 storeData <*h2*> ^ <*y*> by FINSEQ_1:def 9; :: thesis: verum

end;

hence U3(n)Turing computes n proj 3 by Def16; :: thesis: verum