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

theorem
  for s being All-State of ZeroTuring ';' SuccTuring, t be Tape of
ZeroTuring , head,n be Element of NAT st s=[[0,0],head,t] & t storeData <*head,
n*> holds s is Accept-Halt & (Result s)`2_3=head &
  (Result s)`3_3 storeData <*head,
  1*>
proof
  let s be All-State of ZeroTuring ';' SuccTuring, t be Tape of ZeroTuring, h,
  n be Element of NAT;
  assume that
A1: s=[[0,0],h,t] and
A2: t storeData <*h,n*>;
  reconsider h1=h as Element of INT by INT_1:def 2;
  set s1=[the InitS of ZeroTuring,h1,t];
A3: 0=the InitS of ZeroTuring by Def19;
  then
A4: s1 is Accept-Halt & (Result s1)`2_3=h by A2,Lm15;
  the Symbols of ZeroTuring={0,1} by Def19
    .=the Symbols of SuccTuring by Def17;
  then reconsider t2=(Result s1)`3_3 as Tape of SuccTuring;
  set s2=[the InitS of SuccTuring,h1,t2];
A5: 0=the InitS of SuccTuring by Def17;
  then
A6: s=[the InitS of ZeroTuring ';' SuccTuring,h,t] by A1,A3,Def31;
  (Result s1)`3_3 storeData <*h,0 *> by A2,A3,Lm15;
  then
A7: t2 storeData <*h,0*> by Th48;
  then
A8: (Result s2)`3_3 storeData <*h,0+ 1 *> by A5,Th31;
A9: s2 is Accept-Halt by A7,A5,Th31;
  hence s is Accept-Halt by A4,A6,Th45;
  (Result s2)`2_3=h by A7,A5,Th31;
  hence (Result s)`2_3=h by A4,A9,A6,Th45;
  (Result s)`3_3=(Result s2)`3_3 by A4,A9,A6,Th45;
  hence thesis by A8,Th48;
end;
