reserve n,i,j,k for Nat;
reserve T for TuringStr,
  s for All-State of T;

theorem
  SuccTuring computes 1 succ 1
proof
  now
    set sc=1 succ 1;
    let s be All-State of SuccTuring,t be Tape of SuccTuring, h be Element of
    NAT, x be FinSequence of NAT;
    assume that
A1: x in dom sc and
A2: s=[the InitS of SuccTuring,h,t] and
A3: t storeData <*h*>^x;
A4: s = [0,h,t] by A2,Def17;
A5: dom sc = 1-tuples_on NAT by COMPUT_1:def 7;
    then x is Tuple of 1,NAT by A1,FINSEQ_2:131;
    then consider i being Element of NAT such that
A6: x = <*i*> by FINSEQ_2:97;
A7: <*h*>^x=<*h,i*> by A6;
    hence s is Accept-Halt by A3,A4,Th31;
    take h2=h;
    take y=i+1;
    thus (Result s)`2_3=h2 by A3,A4,A7,Th31;
    dom <*i*> = {1} by FINSEQ_1:def 8,FINSEQ_1:2;
    then 1 in dom <*i*> by TARSKI:def 1;
    then
A8: x/.1 = x.1 by A6,PARTFUN1:def 6;
    thus y=(x/.1)+1 by A6,FINSEQ_4:16
    .=sc.x by A1,A5,A8,COMPUT_1:def 7;
    (Result s)`3_3 storeData <*h2,i+1 *> by A3,A4,A7,Th31;
    hence (Result s)`3_3 storeData <*h2*>^<* y *>;
  end;
  hence thesis;
end;
