us to consider the $Z$-part of the flag error set $\mathcal{E}_Z(g)=\{I,Z_{q_1},Z_{q_4},Z_{q_3}Z_{q_4} \}$. As required, the elements of $\mathcal{E}_Z(g)$ all have distinct syndromes (with satisfied stabilizers represented by a plus). []{data-label="fig:Steane"}](SteaneConditionSatisfied.png "fig:"){width="\textwidth"}

A tree diagram for the flag $1$-FTEC Protocol is illustrated in \[fig:TreeD3Diag\]. We now outline the proof that the flag 1-FTEC protocol satisfies the fault-tolerance criteria of \[Def:FaultTolerantDef\] (a more rigorous proof of the general case is presented in \[app:ProtocolGeneralProof\]). To show that Flag $1$-FTEC Protocol satisfies the criteria of \[Def:FaultTolerantDef\], we can assume there is at most one fault during the protocol. If a single fault occurs in either the first or second round leading to a flag, repeating the syndrome measurement will correctly diagnose the error. If there are no flags and a fault occurs which causes the syndromes in the first two rounds to change, then the syndrome during the third round will correctly diagnose the error. There could also be a fault during either the first or second round that goes undetected. But since there were no flags it cannot spread to an error of weight-2. In this case applying a minimum weight correction based on the measured syndrome of the second round will guarantee 