<?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="waybel_3.html#V4">non-Empty</a> <a href="pencil_1.html#V11" target="_self">TopStruct-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="relat_1.html#V2">V3</a> <a href="pencil_1.html#V13" target="_self">trivial-yielding</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>given </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:51"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="pencil_1.html#V3" target="_self">void</a>
 ;<br/>

<a NAME="E3:51"/><i><font color="Green">E35</font></i>: 
not 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#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E2:51"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pencil_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</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#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<a NAME="E5:51"/><i><font color="Green">E36</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> st <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( not <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:51_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E2:51_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</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="E3:51_1"/><b>then </b><i><font color="Green">E38</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/>
<b>assume </b><a NAME="E4:51_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E5:51_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/>
<a NAME="E6:51_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</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="txt" href="pencil_1.html#E2:51_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:51_1"/>
( not <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="pencil_1.html#E3:51_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pencil_1.html#E5:51_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pencil_1.html#D16" target="_self">Def16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:51"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>4</sub></font>,<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:51_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:51_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1"><b>suppose </b></a><a NAME="E1:51_2_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:51_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E2:51_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:51_2_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</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#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_1.html#E3:51"><i><font color="Green">E35</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E5:51_2_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</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>6</sub></font> )
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:51_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><b>thus </b><a NAME="E6:51_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</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>6</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:51_2_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="pencil_1.html#E3:51_2_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="pencil_1.html#E5:51_2_1_1"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_2"><b>suppose </b></a><a NAME="E1:51_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:51_2_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:51_2_1_2"/>
<font color="Maroon">c<sub>3</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/><b>hence </b><a NAME="E4:51_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</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>6</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:51_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="pencil_1.html#E2:51_2_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> as  <a href="pencil_1.html#V16" target="_self">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="txt" href="pencil_1.html#E5:51"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pencil_1.html#D20" target="_self">Def20</a>, <a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E8:51"/>
 <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="E9:51"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is   <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#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E10:51"/><b>then </b>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="pencil_1.html#K4" target="_self">Segre_Blocks</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D22" target="_self">Def22</a></i>;<br/>
<b>hence </b><a NAME="E11:51"/>
not  <a href="pencil_1.html#K5" target="_self">Segre_Product</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pencil_1.html#V3" target="_self">void</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#D4" target="_self">Def4</a></i>;<br/>


</div>
