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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="group_1.html#V6">finite</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>set </b><font color="Maroon">kp</font> = <font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font>;<br/>
<b>set </b><font color="Maroon">k1p</font> = <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>;<br/>
<b>assume </b><a NAME="E1:65"/><i><font color="Green">E22</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#NR2" target="_self">&lt;&gt;</a> <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>
 ;<br/>

<a NAME="E2:65"/><i><font color="Green">E23</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="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:65"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:65"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> or  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> )
 
<b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:65"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>
 <b>by </b><i>, , </i>;<br/>


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