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<span class="kw">let </span><font color="Maroon" title="c6">it1</font>, <font color="Maroon" title="c7">it2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">OAlph</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">q</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c3">tfsm</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">qf</font> &amp; <font color="Maroon" title="c6">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">q</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp;  ex <font color="Olive" title="b1">q</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c3">tfsm</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">qf</font> &amp; <font color="Maroon" title="c7">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">q</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) implies <font color="Maroon" title="c6">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">it2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c8">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">tfsm</font><span class="kw"> such that </span><a NAME="E2:80_1_2"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<font color="Maroon" title="c8">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">qf</font>
 <span class="kw">and </span><br/><a NAME="E3:80_1_2"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<font color="Maroon" title="c6">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">q1</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( for <font color="Olive" title="b1">q</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c3">tfsm</font>  holds <br/>(  not <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">qf</font> or  not <font color="Maroon" title="c7">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">q</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) or <font color="Maroon" title="c6">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">it2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c9">q2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">tfsm</font><span class="kw"> such that </span><a NAME="E5:80_1_2"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<font color="Maroon" title="c9">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">qf</font>
 <span class="kw">and </span><br/><a NAME="E6:80_1_2"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
<font color="Maroon" title="c7">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U3" title="FSM_1:sel.3">OFun</a> of <font color="Maroon" title="c3">tfsm</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">q2</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">it2</font></span><br/>

<a NAME="E7:80_1_2"/>
 <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">tfsm</font> is  <a href="fsm_1.html#V4" title="FSM_1:attr.4">final</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D15" target="_self" title="FSM_1:def.15">Def15</a></span>;<br/>
<a NAME="E8:80_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">q1</font>,<font color="Maroon" title="c9">q2</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_2"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E5:80_1_2"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:80_1_2"/>
<font color="Maroon" title="c6">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">it2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_1_2"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E6:80_1_2"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="fsm_1.html#T36" target="_self" title="FSM_1:th.36">Th36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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