<?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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="pencil_1.html#V11">TopStruct-yielding</a> <a href="pencil_1.html#V14">non-Trivial-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<a NAME="E1:8"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</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="E2:8"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="pencil_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:8"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> <a href="pencil_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> is <a href="realset2.html#V3">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D18">PENCIL_1:def 18</a></i>;<br/>
<a NAME="E4:8"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pencil_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pencil_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K5">Segre_Product</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="pboole.html#M2">Element</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="pencil_3.html#T6" target="_self">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</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>3</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>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</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>3</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>, <a class="ref" href="pencil_2.html#T7">PENCIL_2:7</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<a NAME="E10:8"/>
 <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/>
<a NAME="E11:8"/><b>then </b><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<b>hence </b><a NAME="E12:8"/>
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="pencil_1.html#D21">PENCIL_1:def 21</a></i>;<br/>

<a NAME="E13:8"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>8</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>8</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>3</sub></font> <a href="pencil_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E11:8"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E14:8"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>8</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"><a href="pencil_1.html#K5">Segre_Product</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_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> ; <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="E1:8_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</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>8</sub></font> holds <br/><font color="Maroon">c<sub>10</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>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E4:8_1"/><i><font color="Green">E11</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>3</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="E5:8_1"/><b>then </b><i><font color="Green">E12</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E3:8_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:8_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><a NAME="E2:8_1_1"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:8_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pencil_3.html#E5:8_1"><i><font color="Green">E12</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="pencil_3.html#E4:8_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="pencil_3.html#E1:8_1_1"><i><font color="Green">E13</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:8_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E11:8"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pencil_3.html#E2:8_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pencil_3.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</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>12</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:8_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:8_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="pzfmisc1.html#D1">PZFMISC1:def 1</a></i>;<br/><a NAME="E4:8_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:8_1_1_1_2"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</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>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D17">PBOOLE:def 17</a></i>;<br/><a NAME="E6:8_1_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E7:8_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="pencil_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E8:8_1_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> <a href="pencil_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is <a href="realset2.html#V3">trivial</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#D18">PENCIL_1:def 18</a></i>;<br/><a NAME="E9:8_1_1_1_2"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pencil_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pencil_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> as  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E11:8_1_1_1_2"/>
not <font color="Maroon">c<sub>13</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E12:8_1_1_1_2"/><b>then </b>
not  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pencil_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><a NAME="E13:8_1_1_1_2"/><b>then </b>
not <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="pencil_3.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E14:8_1_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:8_1_1_1_2"><i><font color="Green">E15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:8_1"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E3:8_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pencil_3.html#E5:8_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E8:8_1"/>
 <a href="pencil_1.html#K5">Segre_Product</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pencil_1.html#K4">Segre_Blocks</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pencil_1.html#D23">PENCIL_1:def 23</a></i>;<br/>
<b>hence </b><a NAME="E9:8_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K5">Segre_Product</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E7:8_1"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:8"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>8</sub></font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E13:8"><i><font color="Green">E9</font></i></a>, <a class="ref" href="pencil_2.html#D2">PENCIL_2:def 2</a></i>;<br/>


</div>
