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<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">IAlph</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">OAlph</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">tfsm</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fsm_1.html#L2" target="_self">Mealy-FSM</a> of <font color="Maroon">IAlph</font>,<font color="Maroon">OAlph</font>;<br/>







<b>assume </b><a NAME="E1:64"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font>
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:64"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<b>assume </b><a NAME="E3:64"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font>
 ;<br/>

<b>consider </b><font color="Maroon">d</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:64"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E7:64"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:64"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<b>then consider </b><font color="Maroon">e</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:64"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>thus </b><a NAME="E11:64"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="fsm_1.html#T10" target="_self">Th10</a>, </i>;<br/>


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