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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>







<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>assume </b><a NAME="E1:116"/><i><font color="Green">E86</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="fsm_1.html#R4" target="_self">-are_equivalent</a>  )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="fsm_1.html#K11" target="_self">the_reduction_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E2:116"/><i><font color="Green">E87</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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D18" target="_self">Def18</a></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> ] means ex <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></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">E88</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></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">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:116_1"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font>] )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:116_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:116_1"><i><font color="Green">E89</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:116_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:116_1"><i><font color="Green">E89</font></i></a></i>;<br/>
<b>assume </b><a NAME="E7:116_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<a NAME="E8:116_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E4:116_1"><i><font color="Green">E90</font></i></a></i>;<br/>
<a NAME="E9:116_1"/>
<font color="Maroon">c<sub>16</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E10:116_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></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">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D24" target="_self">Def24</a></i>;<br/>
<a NAME="E11:116_1"/><b>then </b><i><font color="Green">E94</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E8:116_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E12:116_1"/><i><font color="Green">E95</font></i>: 
not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></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">E96</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E4:116_1"><i><font color="Green">E90</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E4:116_1"><i><font color="Green">E90</font></i></a></i>;<br/>
<a NAME="E4:116_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E4:116_1"><i><font color="Green">E90</font></i></a>, <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#E1:116"><i><font color="Green">E86</font></i></a>, <a class="txt" href="fsm_1.html#E1:116_1_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="fsm_1.html#T78" target="_self">Th78</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:116_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E6:116_1"><i><font color="Green">E91</font></i></a></i>;<br/>
<a NAME="E14:116_1"/>
<font color="Maroon">c<sub>17</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E15:116_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></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">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D24" target="_self">Def24</a></i>;<br/>
<a NAME="E16:116_1"/><b>then </b><i><font color="Green">E97</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E13:116_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E17:116_1"/><i><font color="Green">E98</font></i>: 
not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></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">E99</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>17</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E6:116_1"><i><font color="Green">E91</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>17</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E6:116_1"><i><font color="Green">E91</font></i></a></i>;<br/>
<a NAME="E4:116_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E6:116_1"><i><font color="Green">E91</font></i></a>, <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#E1:116"><i><font color="Green">E86</font></i></a>, <a class="txt" href="fsm_1.html#E1:116_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="fsm_1.html#T78" target="_self">Th78</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:116_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>16</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E19:116_1"/>
<font color="Maroon">c<sub>17</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E20:116_1"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> or <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="txt" href="fsm_1.html#E18:116_1"><i><font color="Green">E99</font></i></a>, <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#E1:116"><i><font color="Green">E86</font></i></a>, <a class="txt" href="fsm_1.html#E4:116_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="fsm_1.html#E6:116_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="fsm_1.html#E7:116_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="fsm_1.html#E11:116_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="fsm_1.html#E12:116_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="fsm_1.html#E16:116_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="fsm_1.html#E17:116_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="fsm_1.html#T79" target="_self">Th79</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">E89</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>12</sub></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">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:116_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></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">E91</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D24" target="_self">Def24</a></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:116_2"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:116_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="fsm_1.html#E6:116_2"><i><font color="Green">E92</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>14</sub></font><b> such that </b><br/><a NAME="E9:116_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>14</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:116"><i><font color="Green">E86</font></i></a>, <a class="ref" href="fsm_1.html#T80" target="_self">Th80</a></i>;<br/>
<a NAME="E10:116_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E1:116"><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="txt" href="fsm_1.html#E9:116_2"><i><font color="Green">E93</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">E94</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E6:116_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font>;<br/>
<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>thus </b><a NAME="E12:116_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>17</sub></font>]
 <b>by </b><i><a class="txt" href="fsm_1.html#E6:116_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="fsm_1.html#E11:116_2"><i><font color="Green">E94</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:116"/><i><font color="Green">E90</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="txt" href="fsm_1.html#E3:116"><i><font color="Green">E88</font></i></a>, <a class="txt" href="fsm_1.html#E4:116"><i><font color="Green">E89</font></i></a>);<br/></i>
<a NAME="E7:116"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:116_3_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 ;<br/><b>assume </b><a NAME="E2:116_3_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:116_3_1"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E6:116"><i><font color="Green">E90</font></i></a>, <a class="txt" href="fsm_1.html#E1:116_3_1"><i><font color="Green">E92</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:116_3_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E6:116"><i><font color="Green">E90</font></i></a>, <a class="txt" href="fsm_1.html#E1:116_3_1"><i><font color="Green">E92</font></i></a></i>;<br/><a NAME="E7:116_3_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>15</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E8:116_3_1"/>
<font color="Maroon">c<sub>16</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <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">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="txt" href="fsm_1.html#E7:116_3_1"><i><font color="Green">E96</font></i></a>, <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#E1:116"><i><font color="Green">E86</font></i></a>, <a class="txt" href="fsm_1.html#E6:116"><i><font color="Green">E90</font></i></a>, <a class="txt" href="fsm_1.html#E1:116_3_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="fsm_1.html#E2:116_3_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="fsm_1.html#E4:116_3_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="fsm_1.html#E6:116_3_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="fsm_1.html#T78" target="_self">Th78</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">c<sub>12</sub></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">E92</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></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">E93</font></i>: 
(  not  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> or  not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></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">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> or  not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> )
 </span><b>by </b><i><a class="txt" href="fsm_1.html#E1:116_4_1"><i><font color="Green">E93</font></i></a></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">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></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">E94</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></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">E95</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E6:116"><i><font color="Green">E90</font></i></a>, <a class="txt" href="fsm_1.html#E3:116_4_1_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:116_4_1_1_1"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E6:116"><i><font color="Green">E90</font></i></a>, <a class="txt" href="fsm_1.html#E5:116_4_1_1_1"><i><font color="Green">E95</font></i></a></i>;<br/><a NAME="E8:116_4_1_1_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:116_4_1_1_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="fsm_1.html#E7:116_4_1_1_1"><i><font color="Green">E96</font></i></a></i>;<br/><a NAME="E9:116_4_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <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">E98</font></i>: 
<font color="Maroon">c<sub>15</sub></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">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="fsm_1.html#D24" target="_self">Def24</a></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">E99</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:116_4_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:116_4_1_1_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="fsm_1.html#E7:116_4_1_1_1"><i><font color="Green">E96</font></i></a>, <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#E1:116"><i><font color="Green">E86</font></i></a>, <a class="txt" href="fsm_1.html#E5:116_4_1_1_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="fsm_1.html#E7:116_4_1_1_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="fsm_1.html#E1:116_4_1_1_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="fsm_1.html#T78" target="_self">Th78</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="txt" href="fsm_1.html#E3:116_4_1_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="fsm_1.html#E8:116_4_1_1_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="fsm_1.html#E10:116_4_1_1_1"><i><font color="Green">E98</font></i></a>, <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">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></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">E94</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></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">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:116_4_1_1_2"><i><font color="Green">E94</font></i></a>, <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">E95</font></i>: 
<font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="fsm_1.html#D24" target="_self">Def24</a></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></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">E96</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:116_4_1_1_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="fsm_1.html#E8:116_4_1_1_2"><i><font color="Green">E96</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>15</sub></font><b> such that </b><br/><a NAME="E11:116_4_1_1_2"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>15</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:116"><i><font color="Green">E86</font></i></a>, <a class="ref" href="fsm_1.html#T80" target="_self">Th80</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E13:116_4_1_1_2"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E6:116"><i><font color="Green">E90</font></i></a>, <a class="txt" href="fsm_1.html#E11:116_4_1_1_2"><i><font color="Green">E97</font></i></a></i>;<br/><a NAME="E14:116_4_1_1_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>15</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E15:116_4_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <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">E100</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="txt" href="fsm_1.html#E14:116_4_1_1_2"><i><font color="Green">E99</font></i></a>, <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">E101</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E13:116_4_1_1_2"><i><font color="Green">E98</font></i></a></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">E102</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><a NAME="E2:116_4_1_1_2_1"/>
<font color="Maroon">c<sub>15</sub></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">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E14:116_4_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="fsm_1.html#D24" target="_self">Def24</a></i>;<br/><a NAME="E3:116_4_1_1_2_1"/><b>then </b><i><font color="Green">E103</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E8:116_4_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="fsm_1.html#E13:116_4_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="fsm_1.html#E16:116_4_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="fsm_1.html#E17:116_4_1_1_2"><i><font color="Green">E101</font></i></a>, <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">E104</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:116_4_1_1_2_1_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E8:116_4_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="fsm_1.html#E13:116_4_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="fsm_1.html#E16:116_4_1_1_2"><i><font color="Green">E100</font></i></a>, <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">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E13:116_4_1_1_2"><i><font color="Green">E98</font></i></a>, <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#E1:116"><i><font color="Green">E86</font></i></a>, <a class="txt" href="fsm_1.html#E11:116_4_1_1_2"><i><font color="Green">E97</font></i></a>, <a class="txt" href="fsm_1.html#E1:116_4_1_1_2_1_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="fsm_1.html#E2:116_4_1_1_2_1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="fsm_1.html#T78" target="_self">Th78</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#E1:116"><i><font color="Green">E86</font></i></a>, <a class="txt" href="fsm_1.html#E3:116_4_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="txt" href="fsm_1.html#E8:116_4_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="fsm_1.html#E13:116_4_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="fsm_1.html#E16:116_4_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="fsm_1.html#E1:116_4_1_1_2_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="fsm_1.html#E3:116_4_1_1_2_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="fsm_1.html#T79" target="_self">Th79</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="txt" href="fsm_1.html#E6:116"><i><font color="Green">E90</font></i></a>, <a class="txt" href="fsm_1.html#E3:116_4_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="txt" href="fsm_1.html#E11:116_4_1_1_2"><i><font color="Green">E97</font></i></a>, <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">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="fsm_1.html#E6:116"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<a NAME="E10:116"/>
( <font color="Maroon">c<sub>13</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>13</sub></font> is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E7:116"><i><font color="Green">E91</font></i></a>, <a class="txt" href="fsm_1.html#E8:116"><i><font color="Green">E92</font></i></a>, <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">E93</font></i>: 
<font color="Maroon">c<sub>13</sub></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">E94</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</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>4</sub></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">E95</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>14</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:116_5"/><i><font color="Green">E96</font></i>: 
( 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> <font color="Maroon">c<sub>14</sub></font> &amp; the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:116"><i><font color="Green">E86</font></i></a>, <a class="ref" href="fsm_1.html#T75" target="_self">Th75</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:116_5"/><i><font color="Green">E97</font></i>: 
( 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> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> 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> <font color="Maroon">c<sub>15</sub></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">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E6:116"><i><font color="Green">E90</font></i></a></i>;<br/>
<a NAME="E6:116_5"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:116_5"/>
<font color="Maroon">c<sub>15</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:116_5"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="txt" href="fsm_1.html#E6:116_5"><i><font color="Green">E98</font></i></a>, <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">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> the <a href="fsm_1.html#U2" target="_self">InitS</a> of <font color="Maroon">c<sub>3</sub></font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E3:116_5"><i><font color="Green">E96</font></i></a>, <a class="txt" href="fsm_1.html#E5:116_5"><i><font color="Green">E97</font></i></a>, <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#E1:116"><i><font color="Green">E86</font></i></a>, <a class="txt" href="fsm_1.html#E1:116_5"><i><font color="Green">E95</font></i></a>, <a class="txt" href="fsm_1.html#E3:116_5"><i><font color="Green">E96</font></i></a>, <a class="ref" href="fsm_1.html#T79" target="_self">Th79</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:116"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">c<sub>3</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> holds <br/>( <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<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>1</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>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></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">c<sub>14</sub></font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:116_6"/><i><font color="Green">E95</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D24" target="_self">Def24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E6:116"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="txt" href="fsm_1.html#E1:116_6"><i><font color="Green">E95</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:116_6"/><i><font color="Green">E96</font></i>: 
 <a href="fsm_1.html#K8" target="_self">final_states_partition</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="fsm_1.html#V4" target="_self">final</a>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D15" target="_self">Def15</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:116_6"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E6:116"><i><font color="Green">E90</font></i></a></i>;<br/>
<a NAME="E8:116_6"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="fsm_1.html#R5" target="_self">-are_equivalent</a> 
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="txt" href="fsm_1.html#E5:116_6"><i><font color="Green">E96</font></i></a>, <a class="txt" href="fsm_1.html#E7:116_6"><i><font color="Green">E97</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self">Def14</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>20</sub></font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</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>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>23</sub></font> = the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></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><a class="txt" href="fsm_1.html#E8:116_6"><i><font color="Green">E98</font></i></a>, <a class="ref" href="fsm_1.html#T36" target="_self">Th36</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>24</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E11:116_6"/><i><font color="Green">E99</font></i>: 
( the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font> &amp; the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="txt" href="fsm_1.html#E5:116_6"><i><font color="Green">E96</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E12:116_6"/><i><font color="Green">E100</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">c<sub>3</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></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">E101</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></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#E1:116"><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">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="fsm_1.html#E12:116_6"><i><font color="Green">E100</font></i></a>, <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">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="fsm_1.html#E13:116_6"><i><font color="Green">E101</font></i></a>, <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">E102</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">c<sub>3</sub></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">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<i><font color="Green">E103</font></i>: the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></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">c<sub>3</sub></font> <a href="funct_4.html#K1">+*</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="fsm_1.html#D24" target="_self">Def24</a></i>
<br/>.= 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</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>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="fsm_1.html#E12:116_6"><i><font color="Green">E100</font></i></a>, <a class="txt" href="fsm_1.html#E15:116_6"><i><font color="Green">E102</font></i></a>, <a class="ref" href="funct_4.html#T17">FUNCT_4:17</a></i>
;<br/>
<a NAME="E17:116_6"/><i><font color="Green">E104</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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></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">E105</font></i>: the <a href="fsm_1.html#U1" target="_self">Tran</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></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">c<sub>3</sub></font> <a href="funct_4.html#K1">+*</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="fsm_1.html#D24" target="_self">Def24</a></i>
<br/>.= 
the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="fsm_1.html#E17:116_6"><i><font color="Green">E104</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
;<br/>
<b>consider </b><font color="Maroon">c<sub>25</sub></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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E20:116_6"/><i><font color="Green">E106</font></i>: 
( the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>3</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>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></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">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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><a class="txt" href="fsm_1.html#E6:116"><i><font color="Green">E90</font></i></a></i>;<br/>
<a NAME="E21:116_6"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>24</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <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">E108</font></i>: 
<font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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">c<sub>4</sub></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">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/><a NAME="E2:116_6_4"/>
<font color="Maroon">c<sub>25</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <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">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> or <font color="Maroon">c<sub>24</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E2:116"><i><font color="Green">E87</font></i></a>, <a class="txt" href="fsm_1.html#E21:116_6"><i><font color="Green">E107</font></i></a>, <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#E1:116"><i><font color="Green">E86</font></i></a>, <a class="txt" href="fsm_1.html#E11:116_6"><i><font color="Green">E99</font></i></a>, <a class="txt" href="fsm_1.html#E16:116_6"><i><font color="Green">E103</font></i></a>, <a class="txt" href="fsm_1.html#E18:116_6"><i><font color="Green">E105</font></i></a>, <a class="txt" href="fsm_1.html#E20:116_6"><i><font color="Green">E106</font></i></a>, <a class="txt" href="fsm_1.html#E1:116_6_4"><i><font color="Green">E108</font></i></a>, <a class="ref" href="fsm_1.html#T79" target="_self">Th79</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">c<sub>13</sub></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">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1" target="_self">Tran</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E24:116_6"/><i><font color="Green">E108</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">c<sub>3</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></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">E109</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">c<sub>4</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="fsm_1.html#E24:116_6"><i><font color="Green">E108</font></i></a>, <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="txt" href="fsm_1.html#E13:116_6"><i><font color="Green">E101</font></i></a>, <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">c<sub>3</sub></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">c<sub>4</sub></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">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> = 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_4.html#K1">+*</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="fsm_1.html#E24:116_6"><i><font color="Green">E108</font></i></a>, <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">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="fsm_1.html#D24" target="_self">Def24</a></i>
<br/>.= 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fsm_1.html#K13" target="_self">-Mealy_union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="fsm_1.html#E8:116_6"><i><font color="Green">E98</font></i></a>, <a class="ref" href="fsm_1.html#T36" target="_self">Th36</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_4.html#K1">+*</a> the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="fsm_1.html#D24" target="_self">Def24</a></i>
<br/>.= 
the <a href="fsm_1.html#U3" target="_self">OFun</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="fsm_1.html#E25:116_6"><i><font color="Green">E109</font></i></a>, <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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="fsm_1.html#R7" target="_self">-are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="fsm_1.html#E11:116"><i><font color="Green">E93</font></i></a>, <a class="txt" href="fsm_1.html#E12:116"><i><font color="Green">E94</font></i></a>, <a class="ref" href="fsm_1.html#D19" target="_self">Def19</a></i>;<br/>


</div>
