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

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="pboole.html#M2">Element</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E4:41_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</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>3</sub></font> )
 <b>by </b><i><a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="pencil_1.html#E4:41_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pboole.html#D17">PBOOLE:def 17</a></i>;<br/>
<a NAME="E6:41_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:41_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:41_1_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>3</sub></font> is <a href="realset2.html#V3">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D17" target="_self">Def17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = 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="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>3</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="realset2.html#D5">REALSET2:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E11:41_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="yellow_8.html#D1">YELLOW_8:def 1</a></i>;<br/>
<a NAME="E12:41_1_1"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</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="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E11:41_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self">Th2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:41_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>10</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="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T3" target="_self">Th3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="pencil_1.html#E14:41_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font>;<br/>
<a NAME="E17:41_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <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="41_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:41_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>14</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="41_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:41_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:41_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</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>14</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:41_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pboole.html#D17">PBOOLE:def 17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_1_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:41_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:41_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E3:41_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</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>14</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E4:41_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="pencil_1.html#E1:41_1_1_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E18:41_1_1"/>
<span class="p1"><a href="pzfmisc1.html#K2">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="pzfmisc1.html#K2">}</a></span> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:41_1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>14</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>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:41_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E1:41_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:41_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:41_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>15</sub></font> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D4">YELLOW_6:def 4</a></i>;<br/>
<a NAME="E6:41_1_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
not 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="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E7:41_1_1_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</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>15</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</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>15</sub></font> )
 
<b>by </b><i><a class="txt" href="pencil_1.html#E17:41_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pboole.html#D17">PBOOLE:def 17</a></i>;<br/>
<a NAME="E8:41_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="yellow_6.html#T9">YELLOW_6:9</a></i>;<br/>
<a NAME="E9:41_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span> <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="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E6:41_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E10:41_1_1_2"/><b>then </b>
<span class="p1"><a href="pzfmisc1.html#K2">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="pzfmisc1.html#K2">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</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="pralg_1.html#K10">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pzfmisc1.html#D2">PZFMISC1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E11:41_1_1_2"/>
<span class="p1"><a href="pzfmisc1.html#K2">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="pzfmisc1.html#K2">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</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>14</sub></font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T9">YELLOW_6:9</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1"><a href="pzfmisc1.html#K2">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="pzfmisc1.html#K2">}</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="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<b>thus </b><a NAME="E20:41_1_1"/>
<font color="Maroon">c<sub>14</sub></font> is <a href="pencil_1.html#V16" target="_self">Segre-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_3"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D20" target="_self">PENCIL_1:def 20</a><br/>

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:41_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E2:41_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <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"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</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="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="pzfmisc1.html#D2">PZFMISC1:def 2</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>
;<br/>
<b>hence </b><a NAME="E4:41_1_1_3"/>
( not <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> is <a href="realset1.html#V1">trivial</a> )
 ;<br/>


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

<a NAME="E21:41_1_1"/>
 <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="E22:41_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E23:41_1_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="E24:41_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <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="E25:41_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="pzfmisc1.html#D2">PZFMISC1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_1_4"/>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> is <a href="realset1.html#V1">trivial</a>
 ;<br/><a NAME="E2:41_1_1_4"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a> or ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</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/><b>hence </b><a NAME="E3:41_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_1.html#E14:41_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="pencil_1.html#E25:41_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#T9">ZFMISC_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E27:41_1_1"/>
not <font color="Maroon">c<sub>14</sub></font> is <a href="pencil_1.html#V13" target="_self">trivial-yielding</a>
 <b>by </b><i><a class="txt" href="pencil_1.html#E22:41_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pencil_1.html#D16" target="_self">Def16</a></i>;<br/>

<b>thus </b><a NAME="E28:41_1_1"/>
<font color="Maroon">c<sub>14</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 ;<br/>


</div>
