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<b>let </b><font color="Maroon">I</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">A</font> be  <a href="pencil_1.html#V15">PLS-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">L</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">A</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</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">L</font></span>)</span></span>)</span>;<br/>







<a NAME="E1:12"/>
<font color="Maroon">B</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">A</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">L</font></span>)</span></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">A</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">L</font></span>)</span></span>)</span> ;<br/>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">L</font>
 ;<br/><a NAME="E2:12_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">B1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:12_1"/>
( not <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">B1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">B1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i>, <a class="ref" href="pencil_1.html#T12">PENCIL_1:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:12"/>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D6">PENCIL_1:def 6</a></i>;<br/>
<a NAME="E5:12"/><b>then </b>
not <font color="Maroon">B1</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">S</font> = <font color="Maroon">L</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">B1</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">A</font> <b>by </b><i>, <a class="ref" href="pencil_1.html#T9">PENCIL_1:9</a>, <a class="ref" href="pencil_1.html#D20">PENCIL_1:def 20</a>, <a class="ref" href="pencil_2.html#T7">PENCIL_2:7</a></i>;<br/>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2"/>
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">L</font>
 ;<br/><a NAME="E2:12_2"/><b>then </b>
( not <font color="Maroon">S</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">S</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">S</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">S</font></span>)</span> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:12_2"/>
contradiction
 <b>by </b><i><a class="ref" href="pencil_1.html#D21">PENCIL_1:def 21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:12"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:12"/><b>then </b>
<font color="Maroon">S</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">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E10:12"/><b>then </b>
<font color="Maroon">S</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">S</font></span>)</span> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</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">S</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:12"/>
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">B</font></span>)</span> is   <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">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="pencil_1.html#T24">PENCIL_1:24</a></i>;<br/>


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