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

<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be  <a href="fsm_1.html#V2" target="_self">strict</a>  <a href="fsm_1.html#L2" target="_self">Mealy-FSM</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:77_1_2"/><i><font color="Green">E55</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> ) ) &amp; the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:77_1_2"/><i><font color="Green">E56</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>6</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> ) ) &amp; the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E3:77_1_2"/><i><font color="Green">E57</font></i>: 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>5</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E1:77_1_2"><i><font color="Green">E55</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>6</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E2:77_1_2"><i><font color="Green">E56</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></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">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E2:77_1_2_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E3:77_1_2_1_1"/><i><font color="Green">E59</font></i>: 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:77_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="fsm_1.html#E2:77_1_2_1_1"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E4:77_1_2_1_1"/><i><font color="Green">E60</font></i>: 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:77_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="fsm_1.html#E2:77_1_2_1_1"><i><font color="Green">E58</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E6:77_1_2_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>13</sub></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="E7:77_1_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:77_1_2_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="fsm_1.html#E4:77_1_2_1_1"><i><font color="Green">E60</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="E4:77_1_2_1"/>
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:77_1_2"/><i><font color="Green">E58</font></i>: 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>5</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E1:77_1_2"><i><font color="Green">E55</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>6</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E2:77_1_2"><i><font color="Green">E56</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></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">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E2:77_1_2_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T11" target="_self">Th11</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> = 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="fsm_1.html#E1:77_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="fsm_1.html#E2:77_1_2_2_1"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="fsm_1.html#E2:77_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="fsm_1.html#E2:77_1_2_2_1"><i><font color="Green">E59</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:77_1_2_2"/>
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R1">=</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:77_1_2"/>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:77_1_2_3"/>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E1:77_1_2"><i><font color="Green">E55</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<a NAME="E3:77_1_2_3"/>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:77_1_2"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<a NAME="E5:77_1_2_3"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E1:77_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="fsm_1.html#E2:77_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:77_1_2_3"/>
the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:77_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="fsm_1.html#E2:77_1_2"><i><font color="Green">E56</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="E6:77_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:77_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="fsm_1.html#E2:77_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="fsm_1.html#E3:77_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="fsm_1.html#E4:77_1_2"><i><font color="Green">E58</font></i></a></i>;<br/>


</div>
