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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="pencil_1.html#V15">PLS-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:30"/><i><font color="Green">E29</font></i>: 
for <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> holds <font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="pencil_1.html#V10">strongly_connected</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pencil_2.html#NM2">Collineation</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<a NAME="E2:30"/><i><font color="Green">E30</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E3:30"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E4:30"/><i><font color="Green">E32</font></i>: 
(  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E5:30"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:30"/><i><font color="Green">E34</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E3:30"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pencil_3.html#E4:30"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pencil_3.html#T21" target="_self">Th21</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:30"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</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>9</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> st <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="pencil_1.html#R1">are_collinear</a>  ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:30"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pencil_3.html#E4:30"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pencil_3.html#T21" target="_self">Th21</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pencil_3.html#E1:30"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pencil_2.html#T24">PENCIL_2:24</a></i>;<br/>
<a NAME="E10:30"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="funct_2.html#V3">bijective</a>
 
<b>by </b><i><a class="ref" href="pencil_2.html#D4">PENCIL_2:def 4</a></i>;<br/>
<a NAME="E11:30"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E12:30"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E3:30"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pencil_3.html#T20" target="_self">Th20</a>, <a class="ref" href="funct_1.html#T125">FUNCT_1:125</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E14:30"/><i><font color="Green">E38</font></i>: 
(  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2">PENCIL_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E16:30"/><i><font color="Green">E39</font></i>: 
(  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2">PENCIL_2:def 2</a></i>;<br/>
<a NAME="E17:30"/><i><font color="Green">E40</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E12:30"><i><font color="Green">E37</font></i></a>, <a class="txt" href="pencil_3.html#E14:30"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pencil_3.html#E16:30"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pencil_3.html#T21" target="_self">Th21</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>12</sub></font>;<br/>
<a NAME="E18:30"/><i><font color="Green">E41</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E6:30"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> ;<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="E4:30_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="pencil_3.html#E4:30"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pencil_3.html#E4:30_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pencil_1.html#T14">PENCIL_1:14</a></i>;<br/><b>set </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="pencil_1.html#T25">PENCIL_1:25</a></i>;<br/><a NAME="E8:30_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:30"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pencil_1.html#D21">PENCIL_1:def 21</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:30_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/><a NAME="E11:30_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E12:30_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><a NAME="E13:30_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E10:30_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="pencil_3.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E14:30_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> ;<br/><a NAME="E16:30_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:30"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_3.html#E8:30"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pencil_1.html#D21">PENCIL_1:def 21</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:30_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>24</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/><a NAME="E19:30_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E20:30_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><a NAME="E21:30_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>24</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E18:30_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="pencil_3.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E22:30_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> ;<br/><b>set </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font>;<br/><a NAME="E24:30_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pencil_1.html#T25">PENCIL_1:25</a></i>;<br/><a NAME="E25:30_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pencil_1.html#T25">PENCIL_1:25</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/><a NAME="E27:30_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>25</sub></font> <a href="pencil_1.html#R1">are_collinear</a> 
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:30"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pencil_3.html#E10:30_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="pencil_3.html#E18:30_1"><i><font color="Green">E44</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>25</sub></font> or <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/><div class="add"><a NAME="E2:30_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:30"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:30_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:30"><i><font color="Green">E33</font></i></a>, <a class="txt" href="pencil_3.html#E4:30_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E4:30_1_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E10:30_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:30_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E6:30_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E7:30_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:30_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="pencil_3.html#E4:30_1_1_1"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E8:30_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:30"><i><font color="Green">E30</font></i></a>, <a class="txt" href="pencil_3.html#E11:30"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>29</sub></font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:30_1_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> )
 ;<br/><b>take </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/><b>thus </b><a NAME="E2:30_1_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="pencil_1.html#R1">are_collinear</a>  )
 <b>by </b><i><a class="txt" href="pencil_3.html#E27:30_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="pencil_3.html#E1:30_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="pencil_3.html#E4:30_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="pencil_3.html#E6:30_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="pencil_3.html#E1:30_1_1_1_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>31</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E3:30_1_1_1_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>30</sub></font>
 ;<br/><b>hence </b><a NAME="E4:30_1_1_1_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:30_1_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div><a NAME="E10:30_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>27</sub></font> <a href="pencil_1.html#R1">are_collinear</a> 
 <b>by </b><i><a class="txt" href="pencil_3.html#E7:30_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="pencil_3.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E11:30_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="pencil_1.html#R1">are_collinear</a> 
 <b>by </b><i><a class="ref" href="pencil_3.html#T1" target="_self">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pencil_1.html#T14">PENCIL_1:14</a></i>;<br/><a NAME="E13:30_1_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:30"><i><font color="Green">E29</font></i></a>, <a class="txt" href="pencil_3.html#E4:30"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pencil_3.html#E14:30"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pencil_3.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E14:30_1_1_1"/><i><font color="Green">E53</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <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">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <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">c<sub>6</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">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>30</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_1_2_1"/>
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:30_1_1_1_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:30"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pencil_3.html#E14:30_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="pencil_3.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:30_1_1_1_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><b>hence </b><a NAME="E4:30_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:30_1_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:30_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:30_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="pencil_3.html#E14:30_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="pencil_3.html#E1:30_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E16:30_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:30_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E17:30_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:30"><i><font color="Green">E29</font></i></a>, <a class="txt" href="pencil_3.html#E4:30"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pencil_3.html#D4" target="_self">Def4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>30</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1_1_1_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_1_3_1"/>
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1_3_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:30_1_1_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:30_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:30_1_1_1_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E18:30_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="pencil_3.html#E1:30_1_1_1_3_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_1_3_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:30_1_1_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:30_1_1_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:30_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="pencil_3.html#E16:30_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="pencil_3.html#E1:30_1_1_1_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E4:30_1_1_1_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:30"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pencil_3.html#E1:30_1_1_1_3"><i><font color="Green">E56</font></i></a>, <a class="txt" href="pencil_3.html#E1:30_1_1_1_3_1_2"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E19:30_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:30_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E20:30_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:30"><i><font color="Green">E29</font></i></a>, <a class="txt" href="pencil_3.html#E4:30"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pencil_3.html#E6:30"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pencil_3.html#D4" target="_self">Def4</a></i>;<br/><div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_1_1_4"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">c<sub>30</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:30_1_1_1_4"/><i><font color="Green">E60</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="pencil_1.html#R1">are_collinear</a>  ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</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>30</sub></font> holds <br/><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:30_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="pencil_3.html#E11:30_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="pencil_3.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E4:30_1_1_1_4"/><i><font color="Green">E61</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:30_1_1_1_4"><i><font color="Green">E59</font></i></a>, <a class="txt" href="pencil_3.html#E3:30_1_1_1_4"><i><font color="Green">E60</font></i></a></i>;<br/><a NAME="E5:30_1_1_1_4"/>
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:30"><i><font color="Green">E30</font></i></a>, <a class="txt" href="pencil_3.html#E4:30"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pencil_3.html#E16:30_1_1_1"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E6:30_1_1_1_4"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:30_1_1_1_4"/><i><font color="Green">E63</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>28</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">c<sub>29</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">c<sub>12</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">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>31</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1_1_1_4_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>31</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_1_4_1_1"/>
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>12</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1_4_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:30_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:30"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pencil_3.html#E6:30_1_1_1_4"><i><font color="Green">E62</font></i></a>, <a class="txt" href="pencil_3.html#E7:30_1_1_1_4"><i><font color="Green">E63</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E3:30_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:30"><i><font color="Green">E38</font></i></a></i>;<br/><b>hence </b><a NAME="E4:30_1_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E16:30"><i><font color="Green">E39</font></i></a>, <a class="txt" href="pencil_3.html#E17:30"><i><font color="Green">E40</font></i></a>, <a class="txt" href="pencil_3.html#E1:30_1_1_1_4_1_1_1"><i><font color="Green">E64</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:30_1_1_1_4_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:30_1_1_1_4"><i><font color="Green">E60</font></i></a>, <a class="txt" href="pencil_3.html#E4:30_1_1_1_4"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E3:30_1_1_1_4_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:30"><i><font color="Green">E30</font></i></a>, <a class="txt" href="pencil_3.html#E4:30"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pencil_3.html#E19:30_1_1_1"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E4:30_1_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E16:30"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E5:30_1_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E7:30_1_1_1_4"><i><font color="Green">E63</font></i></a>, <a class="txt" href="pencil_3.html#E2:30_1_1_1_4_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E9:30_1_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E7:30_1_1_1_4"><i><font color="Green">E63</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E10:30_1_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:30"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pencil_3.html#E6:30_1_1_1_4"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E11:30_1_1_1_4"/><b>then </b>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E12:30_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_3.html#E12:30"><i><font color="Green">E37</font></i></a>, <a class="txt" href="pencil_3.html#E14:30"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pencil_3.html#E16:30"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E22:30_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E23:30_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E24:30_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:30"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>thus </b><a NAME="E25:30_1_1_1"/>
<span class="p1">(<span class="default"><a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E13:30_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="pencil_3.html#E17:30_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="pencil_3.html#E20:30_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="pencil_3.html#E21:30_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="pencil_3.html#E23:30_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="pencil_3.html#E24:30_1_1_1"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/><div class="add"><a NAME="E2:30_1_1_2"/><i><font color="Green">E47</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</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">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>28</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1_1_2_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>28</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_2_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>29</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E10:30_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="pencil_3.html#E18:30_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="pencil_3.html#E1:30_1_1_2"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:30"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E4:30_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:30_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E5:30_1_1_2"/>
<span class="p1">(<span class="default"><a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E20:30"/>
 <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R1">=</a>  <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E18:30"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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