<?xml version="1.0"?>
<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="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="E1:32"/><i><font color="Green">E21</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="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E2:32"/>
( <font color="Maroon">c<sub>3</sub></font> is    <a href="pencil_2.html#M1" target="_self">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</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> st <br/>( not <font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pencil_1.html#V2">strong</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pencil_1.html#V1">closed_under_lines</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="struct_0.html#NM2">Subset</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> : ( not <font color="Olive">b<sub>2</sub></font> is <a href="realset1.html#V1">trivial</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pencil_1.html#V2">strong</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pencil_1.html#V1">closed_under_lines</a> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="pencil_2.html#R1" target="_self">are_joinable</a>  ) </span>}</span>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="pencil_2.html#M1" target="_self">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="pencil_2.html#M1" target="_self">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</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="E4:32_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</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>5</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>5</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:32_1"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</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/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:32_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:32_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</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_2.html#E7:32_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</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="txt" href="pencil_2.html#E5:32_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="pencil_2.html#E9:32_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E11:32_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is    <a href="subset_1.html#M1">Element</a> of <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="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:32_1_1"/>
<font color="Maroon">c<sub>5</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="E3:32_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>5</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="txt" href="pencil_2.html#E1:32_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E4:32_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <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>5</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_2.html#E5:32_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="pencil_2.html#E9:32_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="pencil_2.html#E1:32_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:32_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is    <a href="subset_1.html#M1">Element</a> of <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="txt" href="pencil_2.html#E3:32_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="pboole.html#M2">Element</a> of  <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#D17">PBOOLE:def 17</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="pencil_1.html#NM1">Block</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>5</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E14:32_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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>5</sub></font></span>)</span></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM2">Subset</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>5</sub></font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="pzfmisc1.html#K1">}</a></span> as    <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>by </b><i><a class="ref" href="pencil_1.html#T11">PENCIL_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</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>5</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font> as    <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>by </b><i><a class="ref" href="pencil_2.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E18:32_1"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D6">PENCIL_1:def 6</a></i>;<br/>
<a NAME="E19:32_1"/><b>then </b><i><font color="Green">E26</font></i>: 
not <font color="Maroon">c<sub>11</sub></font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T4">PENCIL_1:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as  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>by </b><i><a class="ref" href="pencil_1.html#T9">PENCIL_1:9</a>, <a class="ref" href="pencil_1.html#T10">PENCIL_1:10</a></i>;<br/>
<a NAME="E21:32_1"/>
 <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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E22:32_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>14</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>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E23:32_1"/><b>then </b>
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E19:32_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pencil_1.html#D21">PENCIL_1:def 21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>14</sub></font> as   <a href="pencil_1.html#NM1">Block</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_2.html#E22:32_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pencil_1.html#T24">PENCIL_1:24</a></i>;<br/>
<a NAME="E25:32_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM2">Subset</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/>
<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>thus </b><a NAME="E27:32_1"/><i><font color="Green">E28</font></i>: 
( not <font color="Maroon">c<sub>16</sub></font> is <a href="realset1.html#V1">trivial</a> &amp; <font color="Maroon">c<sub>16</sub></font> is <a href="pencil_1.html#V2">strong</a> &amp; <font color="Maroon">c<sub>16</sub></font> is <a href="pencil_1.html#V1">closed_under_lines</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T20">PENCIL_1:20</a>, <a class="ref" href="pencil_1.html#T21">PENCIL_1:21</a></i>;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</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> : ( not <font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pencil_1.html#V2">strong</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pencil_1.html#V1">closed_under_lines</a> &amp; <font color="Maroon">c<sub>16</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="pencil_2.html#R1" target="_self">are_joinable</a>  ) </span>}</span>  as    <a href="pencil_2.html#M1" target="_self">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pencil_2.html#E1:32"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pencil_2.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E29:32_1"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</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="E30:32_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>18</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="32_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_2_1"/>
( <font color="Maroon">c<sub>18</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>5</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1"><b>suppose </b></a><a NAME="E1:32_1_2_1_1"/>
<font color="Maroon">c<sub>18</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>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:32_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:32_1_2_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="pencil_2.html#E1:32_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pzfmisc1.html#D1">PZFMISC1:def 1</a></i>;<br/><a NAME="E4:32_1_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="pencil_2.html#E5:32_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="pencil_2.html#E9:32_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="pencil_2.html#E1:32_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><b>hence </b><a NAME="E5:32_1_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="pencil_2.html#E3:32_1_2_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_2"><b>suppose </b></a><a NAME="E1:32_1_2_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:32_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</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>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_2.html#E22:32_1"><i><font color="Green">E27</font></i></a></i>;<br/><b>hence </b><a NAME="E3:32_1_2_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="pencil_2.html#E4:32_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="pencil_2.html#E1:32_1_2_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E31:32_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E4:32_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="pencil_2.html#E5:32_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="pencil_2.html#E29:32_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="card_3.html#T38">CARD_3:38</a></i>;<br/>
<a NAME="E32:32_1"/>
<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="pencil_2.html#R1" target="_self">are_joinable</a> 
 
<b>by </b><i><a class="txt" href="pencil_2.html#E27:32_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pencil_2.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E33:32_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>16</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="struct_0.html#NM2">Subset</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> : ( not <font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pencil_1.html#V2">strong</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pencil_1.html#V1">closed_under_lines</a> &amp; <font color="Maroon">c<sub>16</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="pencil_2.html#R1" target="_self">are_joinable</a>  ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="pencil_2.html#E27:32_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E34:32_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</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="E1:32_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:32_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="pencil_2.html#E33:32_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:32_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E31:32_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E36:32_1"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E37:32_1"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D6">PENCIL_1:def 6</a></i>;<br/>
<a NAME="E38:32_1"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E36:32_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E39:32_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="pencil_2.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E40:32_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</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> : ( not <font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pencil_1.html#V2">strong</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pencil_1.html#V1">closed_under_lines</a> &amp; <font color="Maroon">c<sub>16</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="pencil_2.html#R1" target="_self">are_joinable</a>  ) </span>}</span> 
 ;<br/>


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

<b>given </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</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> such that </b><a NAME="E3:32"/><i><font color="Green">E22</font></i>: 
( not <font color="Maroon">c<sub>4</sub></font> is <a href="realset1.html#V1">trivial</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="pencil_1.html#V2">strong</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="pencil_1.html#V1">closed_under_lines</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</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> : ( not <font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pencil_1.html#V2">strong</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pencil_1.html#V1">closed_under_lines</a> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="pencil_2.html#R1" target="_self">are_joinable</a>  ) </span>}</span>  )
 ;<br/>

<b>thus </b><a NAME="E4:32"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="pencil_2.html#M1" target="_self">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pencil_2.html#E1:32"><i><font color="Green">E21</font></i></a>, <a class="txt" href="pencil_2.html#E3:32"><i><font color="Green">E22</font></i></a>, <a class="ref" href="pencil_2.html#T22" target="_self">Th22</a></i>;<br/>


</div>
