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<b>let </b><font color="Maroon">it1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font>, <font color="Maroon">it2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font>;<br/>

<b>given </b><font color="Maroon">q1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm</font>, <font color="Maroon">n1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:75_1_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">qf</font> &amp; <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> &amp; <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> )
 ;<br/>

<b>given </b><font color="Maroon">q2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">tfsm</font>, <font color="Maroon">n2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E2:75_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">qf</font> &amp; <font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> &amp; <font color="Maroon">it2</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q2</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> )
 ;<br/>

<b>set </b><font color="Maroon">m</font> = <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E3:75_1_2"/><i><font color="Green">E42</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="int_1.html#T19">INT_1:19</a></i>;<br/>
<a NAME="E4:75_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
;<br/>
<b>set </b><font color="Maroon">q1'</font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">q2'</font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q2</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<a NAME="E5:75_1_2"/>
 <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">tfsm</font> is <a href="fsm_1.html#V4" target="_self">final</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:75_1_2"/><b>then </b>
<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:75_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:75_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>,the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q2</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, , <a class="txt" href="fsm_1.html#E2:75_1_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E9:75_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">tfsm</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q2</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">n1</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:75_1_2"/>
<font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">it2</font>
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/>


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