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<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">O</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fsm_2.html#L2" target="_self">Moore-SM_Final</a> of <font color="Maroon">I</font>,<font color="Maroon">O</font>;<br/>







<b>assume </b><a NAME="E1:37"/><i><font color="Green">E38</font></i>: 
the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">M</font> <a href="hidden.html#R2">in</a> the <a href="fsm_2.html#U1" target="_self">FinalS</a> of <font color="Maroon">M</font>
 ;<br/>

<b>take </b>
1
; <i><font color="Red">:: according to </font></i><a class="ref" href="fsm_2.html#D8" target="_self">FSM_2:def 8</a><br/>

<b>let </b><font color="Maroon">w</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>;<br/>

<b>assume </b><a NAME="E2:37"/>
<font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<b>thus </b><a NAME="E3:37"/>
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 implies ( the <a href="fsm_1.html#U4">OFun</a> of <font color="Maroon">M</font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U4">OFun</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> the <a href="fsm_2.html#U1" target="_self">FinalS</a> of <font color="Maroon">M</font> ) ) &amp; ( for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 holds <br/> not <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> the <a href="fsm_2.html#U1" target="_self">FinalS</a> of <font color="Maroon">M</font> ) )
 <b>by </b><i>, <a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>


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