<?xml version="1.0"?>
<div class="add">

<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">CRtfsm1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> <a href="fsm_1.html#V5" target="_self">reduced</a> <a href="fsm_1.html#V7" target="_self">connected</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>let </b><font color="Maroon">CRtfsm2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> <a href="fsm_1.html#V5" target="_self">reduced</a> <a href="fsm_1.html#V7" target="_self">connected</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">rtfsm1</font> = <font color="Maroon">CRtfsm1</font>;<br/>
<b>set </b><font color="Maroon">rtfsm2</font> = <font color="Maroon">CRtfsm2</font>;<br/>
<b>assume </b><a NAME="E1:116"/><i><font color="Green">E22</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> <a href="subset_1.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> &amp; <font color="Maroon">CRtfsm1</font>,<font color="Maroon">CRtfsm2</font> <a href="fsm_1.html#R4" target="_self">-are_equivalent</a>  )
 ;<br/>

<b>set </b><font color="Maroon">UN</font> = <font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font>;<br/>
<b>set </b><font color="Maroon">RUN</font> =  <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">SRUN</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">Srtfsm1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font>;<br/>
<b>set </b><font color="Maroon">Srtfsm2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font>;<br/>
<a NAME="E2:116"/><i><font color="Green">E23</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">part</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">part</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">part</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> );<br/>
<a NAME="E3:116"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:116_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y2</font>] )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> ;<br/>
<b>consider </b><font color="Maroon">part1</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:116_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part1</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x'</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">part2</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:116_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part2</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part2</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x'</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E7:116_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y2</font>
 ;<br/>

<a NAME="E8:116_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:116_1"/>
<font color="Maroon">part1</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E10:116_1"/><b>then </b>
<font color="Maroon">part1</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:116_1"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> or <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E12:116_1"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:116_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">part1</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">part1</font> <b>by </b><i/>;<br/>
<a NAME="E4:116_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<b>hence </b><a NAME="E5:116_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E86</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:116_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:116_1"/>
<font color="Maroon">part2</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E15:116_1"/><b>then </b>
<font color="Maroon">part2</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E16:116_1"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> or <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E17:116_1"/><i><font color="Green">E82</font></i>: 
not <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:116_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">y2</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">part2</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">part2</font> <b>by </b><i/>;<br/>
<a NAME="E4:116_1_2"/>
<font color="Maroon">y2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<b>hence </b><a NAME="E5:116_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E86</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:116_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">part1</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E19:116_1"/>
<font color="Maroon">part2</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E20:116_1"/><b>then </b>
( <font color="Maroon">part1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">part2</font> or <font color="Maroon">part1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">part2</font> )
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E21:116_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E86</font></i></a>, , , , , , , , <a class="txt" href="fsm_1.html#E16:116_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:116"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:116_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font>
 ;<br/>

<a NAME="E2:116_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:116_2"/><b>then </b><i><font color="Green">E180</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:116_2"/>
 <a href="tarski.html#K3">union</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">part</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:116_2"/><i><font color="Green">E181</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part</font> &amp; <font color="Maroon">part</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">part</font> = <font color="Maroon">part</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">xs</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">part</font>, <font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">part</font><b> such that </b><br/><a NAME="E9:116_2"/><i><font color="Green">E182</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> &amp; ( for <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">part</font>  holds <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) ) )
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E86</font></i></a>, </i>;<br/>
<a NAME="E10:116_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E86</font></i></a>, <a class="txt" href="fsm_1.html#E1:116_2"><i><font color="Green">E90</font></i></a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E11:116_2"/><b>then </b><i><font color="Green">E183</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<b>set </b><font color="Maroon">IT</font> = <font color="Maroon">y</font>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E12:116_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:116"/><i><font color="Green">E184</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<a NAME="E7:116"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:116_3_1"/><i><font color="Green">E287</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/><b>assume </b><a NAME="E2:116_3_1"/><i><font color="Green">E288</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font>
 ;<br/><b>consider </b><font color="Maroon">part1</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:116_3_1"/><i><font color="Green">E289</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part1</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">part2</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:116_3_1"/><i><font color="Green">E473</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part2</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, </i>;<br/><a NAME="E7:116_3_1"/><i><font color="Green">E474</font></i>: 
<font color="Maroon">part1</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E8:116_3_1"/>
<font color="Maroon">part2</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E9:116_3_1"/><b>then </b>
( <font color="Maroon">part1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">part2</font> or <font color="Maroon">part1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">part2</font> )
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>hence </b><a NAME="E10:116_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E86</font></i></a>, , , , , , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:116_3"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:116"/><i><font color="Green">E475</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:116_4_1"/><i><font color="Green">E476</font></i>: 
(  not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> or  not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:116_4_1_1"/>
(  not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> or  not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_4_1_1_1"><b>suppose </b></a><a NAME="E1:116_4_1_1_1"/>
not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:116_4_1_1_1"/><i><font color="Green">E477</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:116_4_1_1_1"/><i><font color="Green">E478</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">part1</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:116_4_1_1_1"/><i><font color="Green">E479</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part1</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, </i>;<br/><a NAME="E8:116_4_1_1_1"/><i><font color="Green">E480</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E9:116_4_1_1_1"/>
<font color="Maroon">part1</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E10:116_4_1_1_1"/><b>then </b><i><font color="Green">E481</font></i>: 
<font color="Maroon">part1</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font></span>)</span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_4_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:116_4_1_1_1_1"/><i><font color="Green">E482</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font>
 ;<br/><a NAME="E2:116_4_1_1_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/><b>hence </b><a NAME="E3:116_4_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E86</font></i></a>, , , , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:116_4_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_4_1_1_2"><b>suppose </b></a><a NAME="E1:116_4_1_1_2"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:116_4_1_1_2"/><i><font color="Green">E483</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> &amp; not <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:116_4_1_1_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:116_4_1_1_2"/><b>then </b><i><font color="Green">E484</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:116_4_1_1_2"/>
 <a href="tarski.html#K3">union</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:116_4_1_1_2"/><i><font color="Green">E485</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Z</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">q1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>, <font color="Maroon">q2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font><b> such that </b><br/><a NAME="E11:116_4_1_1_2"/><i><font color="Green">E486</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> &amp; ( for <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>  holds <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> or <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> ) ) )
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E86</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">F</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E13:116_4_1_1_2"/><i><font color="Green">E487</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, </i>;<br/><a NAME="E14:116_4_1_1_2"/><i><font color="Green">E488</font></i>: 
<font color="Maroon">Z</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E15:116_4_1_1_2"/>
<font color="Maroon">F</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E16:116_4_1_1_2"/><b>then </b><i><font color="Green">E489</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> or <font color="Maroon">Z</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> )
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><a NAME="E17:116_4_1_1_2"/><i><font color="Green">E490</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_4_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:116_4_1_1_2_1"/><i><font color="Green">E491</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q1</font>
 ;<br/><a NAME="E2:116_4_1_1_2_1"/>
<font color="Maroon">Z</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:116_4_1_1_2_1"/><b>then </b><i><font color="Green">E492</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> )
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_4_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:116_4_1_1_2_1_1"/><i><font color="Green">E493</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font>
 ;<br/><a NAME="E2:116_4_1_1_2_1_1"/><i><font color="Green">E494</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E3:116_4_1_1_2_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/><b>hence </b><a NAME="E4:116_4_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E86</font></i></a>, , <a class="ref" href="fsm_1.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:116_4_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E86</font></i></a>, , , , , , , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E19:116_4_1_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:116_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<a NAME="E10:116"/>
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E11:116"/><b>then </b><i><font color="Green">E495</font></i>: 
<font color="Maroon">f</font> is <a href="funct_2.html#V3">bijective</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E12:116"/><i><font color="Green">E496</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">CRtfsm1</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">CRtfsm2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:116_5"/><i><font color="Green">E497</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">CRtfsm1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">CRtfsm2</font>
 ;<br/>

<b>consider </b><font color="Maroon">Q</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:116_5"/><i><font color="Green">E498</font></i>: 
( the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">CRtfsm1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">CRtfsm2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E86</font></i></a>, <a class="ref" href="fsm_1.html#T15" target="_self">Th15</a></i>;<br/>
<b>consider </b><font color="Maroon">part</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:116_5"/><i><font color="Green">E499</font></i>: 
( the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">CRtfsm1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">CRtfsm1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">CRtfsm1</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E6:116_5"/><i><font color="Green">E500</font></i>: 
<font color="Maroon">Q</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:116_5"/>
<font color="Maroon">part</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:116_5"/><b>then </b>
( <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">part</font> or <font color="Maroon">Q</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">part</font> )
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E9:116_5"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">CRtfsm1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Q</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E10:116_5"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E86</font></i></a>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:116"/>
for <font color="Olive">q</font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">CRtfsm1</font><br/> for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font> holds <br/> ( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">q</font>,<font color="Olive">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm2</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span>,<font color="Olive">s</font> &amp; the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">CRtfsm1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">q</font>,<font color="Olive">s</font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">CRtfsm2</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span>,<font color="Olive">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">CRtfsm1</font>;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">IAlph</font>;<br/>



<a NAME="E1:116_6"/><i><font color="Green">E501</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:116_6"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fq'</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span> <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:116_6"/><i><font color="Green">E502</font></i>: 
 <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span> is <a href="fsm_1.html#V4" target="_self">final</a>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">part</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:116_6"/><i><font color="Green">E503</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">part</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E8:116_6"/><i><font color="Green">E504</font></i>: 
<font color="Maroon">q'</font>,<font color="Maroon">fq'</font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 
<b>by </b><i>, , , </i>;<br/>
<b>set </b><font color="Maroon">qu</font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">qfu</font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">fq'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">q1</font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">fq</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>;<br/>
<b>set </b><font color="Maroon">q2</font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<a NAME="E9:116_6"/>
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q'</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 <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">fq'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">X</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E11:116_6"/><i><font color="Green">E505</font></i>: 
( the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">fq'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E12:116_6"/><i><font color="Green">E506</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:116_6"/><i><font color="Green">E507</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm2</font></span>)</span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T122">ZFMISC_1:122</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>
;<br/>
<a NAME="E15:116_6"/><b>then </b><i><font color="Green">E508</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm1</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<i><font color="Green">E509</font></i>: the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm1</font> <a href="funct_4.html#K1">+*</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T17">FUNCT_4:17</a></i>
;<br/>
<a NAME="E17:116_6"/><i><font color="Green">E510</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E511</font></i>: the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">fq'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm1</font> <a href="funct_4.html#K1">+*</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">fq'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="fsm_1.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
;<br/>
<b>consider </b><font color="Maroon">XX</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p2">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E20:116_6"/><i><font color="Green">E558</font></i>: 
( the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm1</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</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">CRtfsm1</font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E21:116_6"/><i><font color="Green">E559</font></i>: 
<font color="Maroon">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_6_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:116_6_4"/><i><font color="Green">E560</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm1</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm2</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/><a NAME="E2:116_6_4"/>
<font color="Maroon">XX</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E3:116_6_4"/><b>then </b>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XX</font> or <font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">XX</font> )
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>hence </b><a NAME="E4:116_6_4"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E86</font></i></a>, , , , , , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E23:116_6"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">q</font>,<font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">CRtfsm2</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">s</font>
 ;<br/>

<a NAME="E24:116_6"/><i><font color="Green">E561</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">CRtfsm1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E25:116_6"/><i><font color="Green">E562</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">CRtfsm2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">CRtfsm1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">CRtfsm2</font></span>)</span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm1</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">CRtfsm2</font></span>)</span>,<font color="Maroon">IAlph</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T122">ZFMISC_1:122</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>
;<br/>
<a NAME="E27:116_6"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">CRtfsm1</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">CRtfsm2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b>the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">CRtfsm1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">q</font>,<font color="Maroon">s</font> = 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">CRtfsm1</font> <a href="funct_4.html#K1">+*</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">CRtfsm2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T17">FUNCT_4:17</a></i>
<br/>.= 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <span class="p1">(<span class="default"><font color="Maroon">CRtfsm1</font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">CRtfsm2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">fq'</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">CRtfsm1</font> <a href="funct_4.html#K1">+*</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">CRtfsm2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">CRtfsm2</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">s</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:116"/>
<font color="Maroon">CRtfsm1</font>,<font color="Maroon">CRtfsm2</font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a> 
 <b>by </b><i>, , <a class="ref" href="fsm_1.html#T11" target="_self">Th11</a></i>;<br/>


</div>
