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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="E1:38"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">M</font> is <a href="fsm_2.html#V1" target="_self">calculating_type</a> &amp; <font color="Maroon">s</font> <a href="fsm_2.html#R2" target="_self">leads_to_final_state_of</a> <font color="Maroon">M</font> &amp; not 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>then consider </b><font color="Maroon">q</font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:38"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">q</font> <a href="fsm_2.html#R1" target="_self">is_accessible_via</a> <font color="Maroon">s</font> &amp; <font color="Maroon">q</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/>;<br/>
<b>consider </b><font color="Maroon">w</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E5:38"/><i><font color="Green">E40</font></i>: 
the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">M</font>,<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font> <a href="fsm_1.html#R1">-leads_to</a> <font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:38"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font></span>)</span>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="fsm_1.html#D3">FSM_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">m</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> <b> such that </b><br/><a NAME="E8:38"/><i><font color="Green">E46</font></i>: 
( ( for <font color="Olive">w</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>  st <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> &amp; <font color="Olive">w</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Olive">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="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> ) &amp; ( for <font color="Olive">w</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font><br/> for <font color="Olive">j</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">j</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="Olive">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">w</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> holds <br/> not <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Olive">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">j</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>, </i>;<br/>
<a NAME="E9:38"/><i><font color="Green">E47</font></i>: 
( not <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/>
<a NAME="E10:38"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:38"/><b>then </b>
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font></span>)</span>,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="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>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = 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> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font></span>)</span>,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="Maroon">m</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">O</font> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>take </b>
<font color="Maroon">t</font>
;<br/>

<b>take </b>
<font color="Maroon">m</font>
; <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">w1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">I</font>;<br/>

<b>assume </b><a NAME="E13:38"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">w1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E14:38"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">w</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/>
<b>hence </b><a NAME="E15:38"/>
( <font color="Maroon">m</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">w1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 implies ( <font color="Maroon">t</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">w1</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="Maroon">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">w1</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="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> ) )
 <b>by </b><i>, , , , </i>;<br/>

<b>thus </b><a NAME="E16:38"/>
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> <font color="Maroon">m</font> &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">w1</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">w1</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>, </i>;<br/>


</div>
