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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</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>2</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:37"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="abian.html#NR4" target="_self">has_no_fixpoint</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span>] means ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K5">`1</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K6">`2</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K7">`3</a>  );<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> =  <a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:37"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span> 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="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:37_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="abian.html#E2:37_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="abian.html#V1" target="_self">even</a> );<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<a NAME="E4:37_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="abian.html#V1" target="_self">even</a> );<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<a NAME="E6:37_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>9</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K4">]</a></span>;<br/><a NAME="E2:37_1_1_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T47">MCART_1:47</a></i>;<br/><b>thus </b><a NAME="E3:37_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</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="E1:37_1_1_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 </span><b>by </b><i><a class="txt" href="abian.html#E2:37_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E1:37_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:37_1_1_1_1_1_1_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="abian.html#V1" target="_self">even</a> ) )
 ;<br/><b>thus </b><a NAME="E4:37_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:37_1_1_1_1_1_1_2"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="abian.html#V1" target="_self">even</a> ) )
 ;<br/><b>thus </b><a NAME="E4:37_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_1_1_1_1_2"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_1_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_1_1_1_1_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:37_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<a NAME="E4:37_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E2:37_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:37_1_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>15</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="abian.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E7:37_1_1_1_1"/><i><font color="Green">E19</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>15</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_2"/>
( <font color="Maroon">c<sub>14</sub></font> is <a href="abian.html#V1" target="_self">even</a> or  not <font color="Maroon">c<sub>14</sub></font> is <a href="abian.html#V1" target="_self">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="abian.html#V1" target="_self">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as  <a href="abian.html#V1" target="_self">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>thus </b><a NAME="E3:37_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> is <a href="abian.html#V1" target="_self">even</a> or  not <font color="Maroon">c<sub>15</sub></font> is <a href="abian.html#V1" target="_self">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> is <a href="abian.html#V1" target="_self">even</a>
 ;<br/><div class="add"><a NAME="E2:37_1_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E6:37_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E1:37_1_1_1_1_2_1"><i><font color="Green">E20</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 <b>by </b><i><a class="txt" href="abian.html#E2:37_1_1_1"><i><font color="Green">E17</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="37_1_1_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2_1_1_1_2"/>
not <font color="Maroon">c<sub>15</sub></font> is <a href="abian.html#V1" target="_self">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as  <a href="abian.html#V1" target="_self">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>17</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E6:37_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E7:37_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>17</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E7:37_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
;<br/><a NAME="E4:37_1_1_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>hence </b><a NAME="E5:37_1_1_1_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 <b>by </b><i><a class="txt" href="abian.html#E2:37_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2_2"/><i><font color="Green">E20</font></i>: 
not <font color="Maroon">c<sub>14</sub></font> is <a href="abian.html#V1" target="_self">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as  <a href="abian.html#V1" target="_self">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>thus </b><a NAME="E3:37_1_1_1_1_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_2_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_2_2_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> is <a href="abian.html#V1" target="_self">even</a> or  not <font color="Maroon">c<sub>15</sub></font> is <a href="abian.html#V1" target="_self">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> is <a href="abian.html#V1" target="_self">even</a>
 ;<br/><div class="add"><a NAME="E2:37_1_1_1_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E6:37_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E1:37_1_1_1_1_2_2"><i><font color="Green">E20</font></i></a></i>;<br/><b>hence </b><a NAME="E3:37_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 <b>by </b><i><a class="txt" href="abian.html#E2:37_1_1_1"><i><font color="Green">E17</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="37_1_1_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_2_2_1_1_2"/>
not <font color="Maroon">c<sub>15</sub></font> is <a href="abian.html#V1" target="_self">even</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as  <a href="abian.html#V1" target="_self">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>17</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E6:37_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E7:37_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>17</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E7:37_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
;<br/><a NAME="E4:37_1_1_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>hence </b><a NAME="E5:37_1_1_1_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 <b>by </b><i><a class="txt" href="abian.html#E2:37_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E4:37_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K5">`1</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:37_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_2_1"><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1_1_1_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font>
 ;<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="E3:37_1_1_1_2_1"/><i><font color="Green">E18</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>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:37_1_1_1_2_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="abian.html#V1" target="_self">even</a> ) )
 <b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_1_2_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_1_2_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E5:37_1_1_1_2_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="abian.html#T6" target="_self">Th6</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="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:37_1_1_1_2_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>17</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Maroon">c<sub>17</sub></font> is <a href="abian.html#V1" target="_self">even</a> )
 <b>by </b><i><a class="txt" href="abian.html#E5:37_1_1_1_2_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> as  <a href="abian.html#V1" target="_self">even</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="abian.html#E8:37_1_1_1_2_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1" target="_self">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1" target="_self">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1" target="_self">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E13:37_1_1_1_2_1"/><i><font color="Green">E21</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>18</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>19</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>21</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E22</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T71">FUNCT_7:71</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font>) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E13:37_1_1_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E13:37_1_1_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>18</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>19</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E8:37_1_1_1_2_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="abian.html#E13:37_1_1_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>19</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E13:37_1_1_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>21</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>21</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E13:37_1_1_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>22</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
;<br/>
<a NAME="E16:37_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_1_2_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E5:37_1_1_1_2_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="abian.html#E14:37_1_1_1_2_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:37_1_1_1_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:37_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K5">`1</a> 
 <b>by </b><i><a class="txt" href="abian.html#E1:37_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="abian.html#E2:37_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:37_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K6">`2</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:37_1_1_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_3_1"><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1_1_1_3_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>9</sub></font>
 ;<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="E3:37_1_1_1_3_1"/><i><font color="Green">E18</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>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:37_1_1_1_3_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="abian.html#V1" target="_self">even</a> ) )
 <b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_1_3_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_1_3_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E5:37_1_1_1_3_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="abian.html#T6" target="_self">Th6</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="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:37_1_1_1_3_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>17</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>16</sub></font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Maroon">c<sub>17</sub></font> is <a href="abian.html#V1" target="_self">even</a> )
 <b>by </b><i><a class="txt" href="abian.html#E5:37_1_1_1_3_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as  <a href="abian.html#V1" target="_self">odd</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="abian.html#E8:37_1_1_1_3_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> as  <a href="abian.html#V1" target="_self">even</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="abian.html#E8:37_1_1_1_3_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1" target="_self">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1" target="_self">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1" target="_self">even</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E14:37_1_1_1_3_1"/><i><font color="Green">E21</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>18</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>19</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>21</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E22</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T71">FUNCT_7:71</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font>) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E14:37_1_1_1_3_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E14:37_1_1_1_3_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>18</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>19</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E8:37_1_1_1_3_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="abian.html#E14:37_1_1_1_3_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>19</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E14:37_1_1_1_3_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>21</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>21</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E14:37_1_1_1_3_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>22</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
;<br/>
<a NAME="E17:37_1_1_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_1_3_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E5:37_1_1_1_3_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="abian.html#E15:37_1_1_1_3_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:37_1_1_1_3_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:37_1_1_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K6">`2</a> 
 <b>by </b><i><a class="txt" href="abian.html#E1:37_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="abian.html#E2:37_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E6:37_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K7">`3</a> 
 <b>by </b><i><a class="txt" href="abian.html#E2:37_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_2"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:37_1_1_2"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="abian.html#V1" target="_self">even</a> ) )
 <b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_2"><i><font color="Green">E16</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E7:37_1_1_2"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="abian.html#V1" target="_self">even</a> ) )
 <b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_2"><i><font color="Green">E16</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="abian.html#E5:37_1_1_2"><i><font color="Green">E17</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:37_1_1_2"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>15</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Maroon">c<sub>16</sub></font> is <a href="abian.html#V1" target="_self">even</a> )
 <b>by </b><i><a class="txt" href="abian.html#E5:37_1_1_2"><i><font color="Green">E17</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:37_1_1_2"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>17</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>18</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>17</sub></font> is <a href="abian.html#V1" target="_self">even</a> &amp; <font color="Maroon">c<sub>18</sub></font> is <a href="abian.html#V1" target="_self">even</a> )
 <b>by </b><i><a class="txt" href="abian.html#E7:37_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/><a NAME="E13:37_1_1_2"/><i><font color="Green">E21</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>15</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>17</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>18</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><i><font color="Green">E22</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>17</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font>) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>18</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E7:37_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E12:37_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="abian.html#E13:37_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>18</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E13:37_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>18</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
;<br/><i><font color="Green">E23</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>18</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>15</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>18</sub></font>) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E5:37_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E10:37_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="abian.html#E13:37_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>18</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E13:37_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>18</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
;<br/><a NAME="E16:37_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being <a href="abian.html#V1" target="_self">odd</a> <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>18</sub></font> as  <a href="abian.html#V1" target="_self">even</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="abian.html#E10:37_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="abian.html#E12:37_1_1_2"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>17</sub></font> as  <a href="abian.html#V1" target="_self">odd</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="abian.html#E12:37_1_1_2"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E3:37_1_1_2_3"/><i><font color="Green">E24</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_2_3_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_2_3_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="abian.html#E1:37_1_1_2_3_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as  <a href="abian.html#V1" target="_self">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>take </b>
<font color="Maroon">c<sub>25</sub></font>
;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_2_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E14:37_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="abian.html#E15:37_1_1_2"><i><font color="Green">E23</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:37_1_1_2_3_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><b>thus </b><a NAME="E6:37_1_1_2_3_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="abian.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_2_3_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="abian.html#E1:37_1_1_2_3_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as  <a href="abian.html#V1" target="_self">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>take </b>
<font color="Maroon">c<sub>25</sub></font>
;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_2_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E14:37_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="abian.html#E15:37_1_1_2"><i><font color="Green">E23</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:37_1_1_2_3_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><b>thus </b><a NAME="E6:37_1_1_2_3_1_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="abian.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>20</sub></font> being  <a href="abian.html#V1" target="_self">odd</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E18:37_1_1_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="abian.html#E18:37_1_1_2"><i><font color="Green">E24</font></i></a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> </span>}</span> ;<br/><a NAME="E20:37_1_1_2"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</sub></font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>;<br/>
<a NAME="E1:37_1_1_2_4"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>hence </b><a NAME="E2:37_1_1_2_4"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>22</sub></font>) <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>3</sub></font>, <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E22:37_1_1_2"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8">FUNCT_2:sch 8</a>(<a class="txt" href="abian.html#E20:37_1_1_2"><i><font color="Green">E25</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1_1_2_5"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:37_1_1_2_5"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="abian.html#E22:37_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_1_1_2_5"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><a NAME="E6:37_1_1_2_5"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E5:37_1_1_2_5"><i><font color="Green">E28</font></i></a>, <a class="ref" href="eqrel_1.html#T30">EQREL_1:30</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>26</sub></font>, <font color="Maroon">c<sub>27</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:37_1_1_2_5"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="abian.html#D7" target="_self">Def7</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>;<br/><a NAME="E9:37_1_1_2_5"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 <b>by </b><i><a class="ref" href="abian.html#T3" target="_self">Th3</a></i>;<br/><div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>28</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:37_1_1_2_5_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>28</sub></font>]
 ;<br/><a NAME="E2:37_1_1_2_5_1"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> = 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K2">*</a> 1</span>)</span>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>28</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E18:37_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="abian.html#E1:37_1_1_2_5_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="abian.html#E2:37_1_1_2_5_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E4:37_1_1_2_5_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div><a NAME="E11:37_1_1_2_5"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="abian.html#E9:37_1_1_2_5"><i><font color="Green">E30</font></i></a>, <a class="txt" href="abian.html#E10:37_1_1_2_5"><i><font color="Green">E31</font></i></a>);<br/></i><a NAME="E12:37_1_1_2_5"/>
2 <a href="nat_1.html#K2">*</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><a NAME="E13:37_1_1_2_5"/><b>then </b><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><b>set </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>20</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>20</sub></font>;<br/><a NAME="E14:37_1_1_2_5"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="abian.html#E13:37_1_1_2_5"><i><font color="Green">E33</font></i></a>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/><a NAME="E15:37_1_1_2_5"/>
<font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E13:37_1_1_2_5"><i><font color="Green">E33</font></i></a>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>20</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E17:37_1_1_2_5"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="abian.html#E14:37_1_1_2_5"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E18:37_1_1_2_5"/><i><font color="Green">E36</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>26</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>27</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>30</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>26</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>30</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>26</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>30</sub></font>) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>27</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="abian.html#E8:37_1_1_2_5"><i><font color="Green">E29</font></i></a>, <a class="txt" href="abian.html#E18:37_1_1_2_5"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>30</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>27</sub></font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E18:37_1_1_2_5"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E17:37_1_1_2_5"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_7.html#T79">FUNCT_7:79</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E11:37_1_1_2_5"><i><font color="Green">E32</font></i></a></i>
;<br/><a NAME="E20:37_1_1_2_5"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_2_5"><i><font color="Green">E27</font></i></a></i>;<br/><b>hence </b><a NAME="E21:37_1_1_2_5"/>
not <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as  <a href="relat_1.html#V2">non-empty</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>3</sub></font>, <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  =  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>3</sub></font>, <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E26:37_1_1_2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="abian.html#V1" target="_self">even</a>;<br/><b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1">set</a> ] means not <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="abian.html#V1" target="_self">even</a>;<br/><b>set </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <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>21</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/><a NAME="E27:37_1_1_2"/><i><font color="Green">E28</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i><a NAME="E28:37_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <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>21</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E27:37_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <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>21</sub></font></span><a href="domain_1.html#K6">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E30:37_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i><b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>30</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>take </b><font color="Maroon">c<sub>31</sub></font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="domain_1.html#K4">]</a></span>;<br/><a NAME="E33:37_1_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T47">MCART_1:47</a></i>;<br/><a NAME="E34:37_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(0) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="abian.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E35:37_1_1_2"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> </span>}</span> 
 ;<br/><a NAME="E36:37_1_1_2"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E22:37_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E37:37_1_1_2"/><b>then </b>
 <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="abian.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E38:37_1_1_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 0
 <b>by </b><i><a class="txt" href="abian.html#E26:37_1_1_2"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E39:37_1_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/><i><font color="Green">E32</font></i>: <font color="Maroon">c<sub>28</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>30</sub></font> = 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>30</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
<b>by </b><i><a class="txt" href="abian.html#E39:37_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/><a NAME="E41:37_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_7"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>32</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1_1_2_7_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_2_7_1_1"/>
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  or <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font> )
 </span><b>by </b><i><a class="txt" href="abian.html#E1:37_1_1_2_7_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_7_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_2_7_1_1_1"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>33</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:37_1_1_2_7_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font> is <a href="abian.html#V1" target="_self">even</a> )
 ;<br/><b>thus </b><a NAME="E4:37_1_1_2_7_1_1_1"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_2_7_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_7_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_2_7_1_1_2"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>33</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:37_1_1_2_7_1_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> &amp; not <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font> is <a href="abian.html#V1" target="_self">even</a> )
 ;<br/><b>thus </b><a NAME="E4:37_1_1_2_7_1_1_2"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_2_7_1_1_2"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>32</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:37_1_1_2_7"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>32</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<a NAME="E4:37_1_1_2_7"/>
( <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font> is <a href="abian.html#V1" target="_self">even</a> or  not <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font> is <a href="abian.html#V1" target="_self">even</a> )
 
;<br/>
<a NAME="E5:37_1_1_2_7"/><b>then </b>
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  or <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E6:37_1_1_2_7"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E42:37_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E33:37_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="abian.html#E40:37_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><a NAME="E43:37_1_1_2"/><i><font color="Green">E33</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>29</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_1_2_8"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>32</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_2_8"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:37_1_1_2_8"/>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <br/>( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1" target="_self">even</a> ) &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <br/>( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1" target="_self">even</a> ) )
 
<b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_2_8"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:37_1_1_2_8"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E44:37_1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E43:37_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E45:37_1_1_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>21</sub></font> holds <br/><font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>32</sub></font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E1:37_1_1_2_9"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/>

<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1_1_2_9_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:37_1_1_2_9_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="abian.html#E26:37_1_1_2"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E3:37_1_1_2_9_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/><a NAME="E4:37_1_1_2_9_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="abian.html#E22:37_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E5:37_1_1_2_9_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><b>hence </b><a NAME="E6:37_1_1_2_9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="abian.html#E1:37_1_1_2_9"><i><font color="Green">E36</font></i></a>, <a class="ref" href="abian.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="abian.html#T6" target="_self">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E5:37_1_1_2_9"/>
<font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="abian.html#E2:37_1_1_2_9"><i><font color="Green">E37</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>35</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E7:37_1_1_2_9"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E39</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>36</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>36</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>36</sub></font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T71">FUNCT_7:71</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>35</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font>

;<br/>
<a NAME="E9:37_1_1_2_9"/>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E26:37_1_1_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E10:37_1_1_2_9"/><b>then </b>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<a NAME="E11:37_1_1_2_9"/><b>then </b>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="abian.html#E22:37_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E12:37_1_1_2_9"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 
;<br/>
<a NAME="E13:37_1_1_2_9"/><b>then </b>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="abian.html#E8:37_1_1_2_9"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E14:37_1_1_2_9"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E22:37_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E15:37_1_1_2_9"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E26:37_1_1_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E16:37_1_1_2_9"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="abian.html#E14:37_1_1_2_9"><i><font color="Green">E40</font></i></a>, <a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<i><font color="Green">E42</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>34</sub></font> <a href="nat_1.html#K1">+</a> 1) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>34</sub></font>) <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T71">FUNCT_7:71</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>34</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E7:37_1_1_2_9"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E18:37_1_1_2_9"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E26:37_1_1_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E19:37_1_1_2_9"/><b>then </b>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<a NAME="E20:37_1_1_2_9"/><b>then </b>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="abian.html#E22:37_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E21:37_1_1_2_9"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 
;<br/>
<a NAME="E22:37_1_1_2_9"/><b>then </b>
<font color="Maroon">c<sub>34</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="abian.html#E17:37_1_1_2_9"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E23:37_1_1_2_9"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E22:37_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E24:37_1_1_2_9"/>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E26:37_1_1_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E25:37_1_1_2_9"/><b>then </b>
<font color="Maroon">c<sub>34</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>35</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E23:37_1_1_2_9"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<a NAME="E26:37_1_1_2_9"/><b>then </b>
<font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>35</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E27:37_1_1_2_9"/>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="abian.html#E16:37_1_1_2_9"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E46:37_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K5">`1</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_10"><b>proof </b></a><div class="add">

<a NAME="E1:37_1_1_2_10"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>28</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>29</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>32</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1_1_2_10_1"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>28</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>33</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_2_10_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:37_1_1_2_10_1"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_2_10_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>34</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:37_1_1_2_10_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>34</sub></font> is <a href="abian.html#V1" target="_self">even</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>34</sub></font> as  <a href="abian.html#V1" target="_self">even</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="abian.html#E6:37_1_1_2_10_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>32</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_2_10_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="abian.html#E6:37_1_1_2_10_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="abian.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E9:37_1_1_2_10_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>34</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_2_10_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="abian.html#E6:37_1_1_2_10_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<a NAME="E10:37_1_1_2_10_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="abian.html#E45:37_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="abian.html#E3:37_1_1_2_10_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="abian.html#E6:37_1_1_2_10_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:37_1_1_2_10_1"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:37_1_1_2_10"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K5">`1</a> 
 <b>by </b><i><a class="txt" href="abian.html#E33:37_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="abian.html#E44:37_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E47:37_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K6">`2</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_11"><b>proof </b></a><div class="add">

<a NAME="E1:37_1_1_2_11"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>29</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>32</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1_1_2_11_1"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>33</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_2_11_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>34</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:37_1_1_2_11_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font> &amp; not <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>34</sub></font> is <a href="abian.html#V1" target="_self">even</a> )
 <b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_2_11_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>34</sub></font> as  <a href="abian.html#V1" target="_self">odd</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="abian.html#E5:37_1_1_2_11_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>32</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="abian.html#E3:37_1_1_2_11_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="abian.html#E5:37_1_1_2_11_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="abian.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E8:37_1_1_2_11_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="abian.html#E38:37_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="abian.html#E45:37_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="abian.html#E3:37_1_1_2_11_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="abian.html#E5:37_1_1_2_11_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:37_1_1_2_11_1"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:37_1_1_2_11"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K6">`2</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K6">`2</a> 
 <b>by </b><i><a class="txt" href="abian.html#E33:37_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="abian.html#E43:37_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E48:37_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K7">`3</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_1_2_12"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K7">`3</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>32</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_2_12"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K7">`3</a> </span>)</span> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>31</sub></font> <a href="mcart_1.html#K7">`3</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:37_1_1_2_12"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E33:37_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="abian.html#E3:37_1_1_2_12"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>33</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:37_1_1_2_12"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="abian.html#E33:37_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="abian.html#E3:37_1_1_2_12"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E7:37_1_1_2_12"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E6:37_1_1_2_12"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:37_1_1_2_12"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="abian.html#R2" target="_self">is_a_fixpoint_of</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:37_1_1_2_12"><i><font color="Green">E37</font></i></a>, <a class="txt" href="abian.html#E6:37_1_1_2_12"><i><font color="Green">E38</font></i></a>, <a class="ref" href="abian.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E9:37_1_1_2_12"/>
contradiction
 <b>by </b><i><a class="txt" href="abian.html#E1:37"><i><font color="Green">E13</font></i></a>, <a class="ref" href="abian.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span><b> such that </b><br/><a NAME="E4:37"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="abian.html#E2:37"><i><font color="Green">E14</font></i></a>);<br/></i>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> ;<br/>
<b>take </b>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
;<br/>

<b>take </b>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
;<br/>

<b>take </b>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
;<br/>

<b>thus </b><a NAME="E5:37"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span>
 ;<br/><a NAME="E2:37_2_1"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="abian.html#E2:37_2_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1"><b>suppose </b></a><a NAME="E1:37_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_2_1_1_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:37_2_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> 
 <b>by </b><i><a class="txt" href="abian.html#E3:37_2_1_1_1"><i><font color="Green">E17</font></i></a></i>;<br/><b>thus </b><a NAME="E6:37_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E3:37_2_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E5:37_2_1_1_1"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_2"><b>suppose </b></a><a NAME="E1:37_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_2_1_1_2"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:37_2_1_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> 
 <b>by </b><i><a class="txt" href="abian.html#E3:37_2_1_1_2"><i><font color="Green">E17</font></i></a></i>;<br/><b>thus </b><a NAME="E6:37_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E3:37_2_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E5:37_2_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_3"><b>suppose </b></a><a NAME="E1:37_2_1_1_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_2_1_1_3"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:37_2_1_1_3"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> 
 <b>by </b><i><a class="txt" href="abian.html#E3:37_2_1_1_3"><i><font color="Green">E17</font></i></a></i>;<br/><b>thus </b><a NAME="E6:37_2_1_1_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="abian.html#E3:37_2_1_1_3"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E5:37_2_1_1_3"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:37_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>;<br/>
<a NAME="E3:37_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E2:37_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="abian.html#E2:37_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E5:37_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E4:37"><i><font color="Green">E15</font></i></a>, <a class="txt" href="abian.html#E3:37_2"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E6:37_2"/><b>then </b><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a>  or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a>  or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a>  )
 </span><b>by </b><i><a class="txt" href="abian.html#E6:37_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_2_1"><b>suppose </b></a><a NAME="E1:37_2_2_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> 
 ;<br/><div class="add"><a NAME="E2:37_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 ;<br/><a NAME="E3:37_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="abian.html#E1:37_2_2_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E4:37_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:37_2_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_2_2"><b>suppose </b></a><a NAME="E1:37_2_2_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> 
 ;<br/><div class="add"><a NAME="E2:37_2_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 ;<br/><a NAME="E3:37_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="abian.html#E1:37_2_2_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E4:37_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:37_2_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_2_3"><b>suppose </b></a><a NAME="E1:37_2_2_3"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> 
 ;<br/><div class="add"><a NAME="E2:37_2_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 ;<br/><a NAME="E3:37_2_2_3"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="abian.html#E1:37_2_2_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E4:37_2_2_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E6:37"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_3"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_3"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_3"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="abian.html#E3:37_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:37_3"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> 
 <b>by </b><i><a class="txt" href="abian.html#E5:37_3"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>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="E9:37_3"/><i><font color="Green">E19</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>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="abian.html#E3:37_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</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="E11:37_3"/><i><font color="Green">E20</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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="abian.html#E9:37_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E13:37_3"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> 
 <b>by </b><i><a class="txt" href="abian.html#E11:37_3"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E14:37_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E5:37_3"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_3"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E15:37_3"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E16:37_3"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:37"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:37_3"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E19:37_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E16:37_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="abian.html#E18:37_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E20:37_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E11:37_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="abian.html#E13:37_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E21:37_3"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E22:37_3"/><b>then </b><i><font color="Green">E25</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="abian.html#E4:37"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:37_3"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E25:37_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E22:37_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="abian.html#E24:37_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E26:37_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E27:37_3"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E5:37_3"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_3"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E9:37_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_7.html#T72">FUNCT_7:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E27:37_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font>,0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T70">FUNCT_7:70</a></i>
;<br/>
<a NAME="E29:37_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E5:37_3"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E11:37_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="abian.html#E13:37_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="abian.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E30:37_3"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E5:37_3"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_3"><i><font color="Green">E18</font></i></a>, <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/>
<a NAME="E31:37_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</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>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E9:37_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="abian.html#E11:37_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E32:37_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:37"><i><font color="Green">E15</font></i></a>, <a class="txt" href="abian.html#E13:37_3"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E33:37_3"/>
contradiction
 <b>by </b><i><a class="txt" href="abian.html#E5:37_3"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E9:37_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="abian.html#E13:37_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="abian.html#E19:37_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="abian.html#E25:37_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="abian.html#E30:37_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="abian.html#E31:37_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:37"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_4"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_4"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_4"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="abian.html#E3:37_4"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:37_4"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> 
 <b>by </b><i><a class="txt" href="abian.html#E5:37_4"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>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="E9:37_4"/><i><font color="Green">E19</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>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="abian.html#E3:37_4"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</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="E11:37_4"/><i><font color="Green">E20</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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="abian.html#E9:37_4"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E13:37_4"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> 
 <b>by </b><i><a class="txt" href="abian.html#E11:37_4"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E14:37_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E5:37_4"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E15:37_4"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E16:37_4"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:37"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:37_4"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E19:37_4"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E16:37_4"><i><font color="Green">E22</font></i></a>, <a class="txt" href="abian.html#E18:37_4"><i><font color="Green">E23</font></i></a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E20:37_4"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E11:37_4"><i><font color="Green">E20</font></i></a>, <a class="txt" href="abian.html#E13:37_4"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E21:37_4"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E22:37_4"/><b>then </b><i><font color="Green">E25</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="abian.html#E4:37"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:37_4"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E25:37_4"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E22:37_4"><i><font color="Green">E25</font></i></a>, <a class="txt" href="abian.html#E24:37_4"><i><font color="Green">E26</font></i></a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E26:37_4"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E27:37_4"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E5:37_4"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E9:37_4"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_7.html#T72">FUNCT_7:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E27:37_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font>,0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T70">FUNCT_7:70</a></i>
;<br/>
<a NAME="E29:37_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E5:37_4"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_4"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E11:37_4"><i><font color="Green">E20</font></i></a>, <a class="txt" href="abian.html#E13:37_4"><i><font color="Green">E21</font></i></a>, <a class="ref" href="abian.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E30:37_4"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E5:37_4"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/>
<a NAME="E31:37_4"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</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>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E9:37_4"><i><font color="Green">E19</font></i></a>, <a class="txt" href="abian.html#E11:37_4"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E32:37_4"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:37"><i><font color="Green">E15</font></i></a>, <a class="txt" href="abian.html#E13:37_4"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E33:37_4"/>
contradiction
 <b>by </b><i><a class="txt" href="abian.html#E5:37_4"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_4"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E9:37_4"><i><font color="Green">E19</font></i></a>, <a class="txt" href="abian.html#E13:37_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="abian.html#E19:37_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="abian.html#E25:37_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="abian.html#E30:37_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="abian.html#E31:37_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E8:37"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_5"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_5"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span> </span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_5"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="abian.html#E3:37_5"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:37_5"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> 
 <b>by </b><i><a class="txt" href="abian.html#E5:37_5"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>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="E9:37_5"/><i><font color="Green">E19</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>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="abian.html#E3:37_5"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</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="E11:37_5"/><i><font color="Green">E20</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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="abian.html#E9:37_5"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E13:37_5"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> 
 <b>by </b><i><a class="txt" href="abian.html#E11:37_5"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E14:37_5"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E5:37_5"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E15:37_5"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:37"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:37_5"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E18:37_5"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E15:37_5"><i><font color="Green">E22</font></i></a>, <a class="txt" href="abian.html#E17:37_5"><i><font color="Green">E23</font></i></a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E19:37_5"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E11:37_5"><i><font color="Green">E20</font></i></a>, <a class="txt" href="abian.html#E13:37_5"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E20:37_5"/><b>then </b><i><font color="Green">E25</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="abian.html#E4:37"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E22:37_5"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E23:37_5"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E20:37_5"><i><font color="Green">E25</font></i></a>, <a class="txt" href="abian.html#E22:37_5"><i><font color="Green">E26</font></i></a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E24:37_5"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E25:37_5"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="abian.html#E5:37_5"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E9:37_5"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_7.html#T72">FUNCT_7:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="abian.html#E25:37_5"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font>,0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T70">FUNCT_7:70</a></i>
;<br/>
<a NAME="E27:37_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E5:37_5"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_5"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E11:37_5"><i><font color="Green">E20</font></i></a>, <a class="txt" href="abian.html#E13:37_5"><i><font color="Green">E21</font></i></a>, <a class="ref" href="abian.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E28:37_5"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="abian.html#K2" target="_self">=_</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E5:37_5"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>;<br/>
<a NAME="E29:37_5"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</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>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E9:37_5"><i><font color="Green">E19</font></i></a>, <a class="txt" href="abian.html#E11:37_5"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E30:37_5"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="abian.html#E4:37"><i><font color="Green">E15</font></i></a>, <a class="txt" href="abian.html#E13:37_5"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E31:37_5"/>
contradiction
 <b>by </b><i><a class="txt" href="abian.html#E5:37_5"><i><font color="Green">E17</font></i></a>, <a class="txt" href="abian.html#E7:37_5"><i><font color="Green">E18</font></i></a>, <a class="txt" href="abian.html#E9:37_5"><i><font color="Green">E19</font></i></a>, <a class="txt" href="abian.html#E13:37_5"><i><font color="Green">E21</font></i></a>, <a class="txt" href="abian.html#E18:37_5"><i><font color="Green">E24</font></i></a>, <a class="txt" href="abian.html#E23:37_5"><i><font color="Green">E27</font></i></a>, <a class="txt" href="abian.html#E28:37_5"><i><font color="Green">E29</font></i></a>, <a class="txt" href="abian.html#E29:37_5"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>


</div>
