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<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">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">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">O</font>;<br/>







<b>set </b><font color="Maroon">M</font> = <font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font>;<br/>
<b>reconsider </b><font color="Maroon">01</font> = 1 as    <a href="subset_1.html#M2">Element</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E2:48"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">s</font> <a href="fsm_2.html#R2" target="_self">leads_to_final_state_of</a> <font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:48"/><i><font color="Green">E39</font></i>: 
( the <a href="fsm_1.html#U2">InitS</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; the <a href="fsm_2.html#U1" target="_self">FinalS</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> &amp; not 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> &amp; the <a href="fsm_1.html#U4">OFun</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span> <a href="funct_2.html#R1">=</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then 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="E5:48"/><i><font color="Green">E40</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 <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span></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 <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E6:48"/><i><font color="Green">E44</font></i>: 
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 <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span></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 <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m</font>;<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="E1:48_1"/><i><font color="Green">E46</font></i>: 
<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/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:48_1_1"/>
<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">w</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:48_1_1"/><b>then </b>
<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 <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span></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 <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:48_1_1"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U4">OFun</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span> <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 <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span></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">w</font>,the <a href="fsm_1.html#U2">InitS</a> of <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span></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 <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E3:48_1"/>
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">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 <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span></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 <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E8:48"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">01</font> <a href="fsm_2.html#R3" target="_self">is_result_of</a> <font color="Maroon">s</font>,<font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:48"/>
 <a href="fsm_2.html#K2" target="_self">Result</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="fsm_2.html#K1" target="_self">-TwoStatesMooreSM</a> 0,1,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i/>;<br/>


</div>
