<?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>assume </b><a NAME="E1:54"/><i><font color="Green">E37</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 <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="pencil_1.html#V6" target="_self">identifying_close_blocks</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="pencil_1.html#V5" target="_self">with_non_trivial_blocks</a> &amp; not for <font color="Olive">b<sub>2</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#K1" target="_self">.</a> <font color="Olive">b<sub>2</sub></font> is <a href="pencil_1.html#V3" target="_self">void</a> )
 ;<br/>

<a NAME="E2:54"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K5" target="_self">Segre_Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st 2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">b<sub>2</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K5" target="_self">Segre_Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<a NAME="E1:54_1"/>
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="txt" href="pencil_1.html#E1:54"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pencil_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E2:54_1"/><b>then </b>
not  <a href="pencil_1.html#K4" target="_self">Segre_Blocks</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being  <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> such that </b><br/><a NAME="E4:54_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>3</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; ex <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>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</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="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D22" target="_self">Def22</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:54_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</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>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_1.html#E4:54_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E7:54_1"/><i><font color="Green">E40</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="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> holds <br/>( not <font color="Maroon">c<sub>5</sub></font> <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; <font color="Maroon">c<sub>5</sub></font> <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="54_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</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="E1:54_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:54_1_1"/><i><font color="Green">E42</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="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> holds <br/>( not <font color="Maroon">c<sub>5</sub></font> <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; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E4:54_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="pencil_1.html#V5" target="_self">with_non_trivial_blocks</a>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E1:54"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E5:54_1_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>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E6:54_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pencil_1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E6:54_1_1"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font> <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="ref" href="pencil_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E7:54_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E3:54_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:54_1_1"/>
( not <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:54_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="pencil_1.html#E3:54_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:54_1"/>
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="txt" href="pencil_1.html#E1:54"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pencil_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E9:54_1"/><b>then </b>
not  <a href="pencil_1.html#K4" target="_self">Segre_Blocks</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being  <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> such that </b><br/><a NAME="E11:54_1"/><i><font color="Green">E41</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>7</sub></font> &amp; ex <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>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</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="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D22" target="_self">Def22</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:54_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</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>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_1.html#E11:54_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E14:54_1"/><i><font color="Green">E43</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>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( not <font color="Maroon">c<sub>7</sub></font> <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; <font color="Maroon">c<sub>7</sub></font> <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="54_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:54_1_2"/><i><font color="Green">E44</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>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:54_1_2"/><i><font color="Green">E45</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>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( not <font color="Maroon">c<sub>7</sub></font> <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; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E4:54_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> is <a href="pencil_1.html#V5" target="_self">with_non_trivial_blocks</a>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E1:54"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E5:54_1_2"/><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>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E13:54_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="pencil_1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E6:54_1_2"/><b>then </b>
not <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E7:54_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E3:54_1_2"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:54_1_2"/>
( not <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:54_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="pencil_1.html#E3:54_1_2"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>assume </b><a NAME="E15:54_1"/>
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="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:54_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</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#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E18:54_1"/><i><font color="Green">E45</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> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E4:54_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pencil_1.html#E17:54_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="card_3.html#T64">CARD_3:64</a></i>;<br/>
<a NAME="E20:54_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>10</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="txt" href="pencil_1.html#E4:54_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pencil_1.html#E17:54_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="card_3.html#T64">CARD_3:64</a></i>;<br/>
<i><font color="Green">E47</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="pencil_1.html#E18:54_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
<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>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</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="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green">E48</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="pencil_1.html#E20:54_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
<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>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</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="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E27:54_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> <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#E17:54_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="pencil_1.html#E22:54_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="pencil_1.html#E24:54_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</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_1.html#E27:54_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E29:54_1"/>
 <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/>
<a NAME="E30:54_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</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>16</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E18:54_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E31:54_1"/>
 <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/>
<a NAME="E32:54_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</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>16</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E20:54_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E33:54_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>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E27:54_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="pencil_1.html#E30:54_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E34:54_1"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E35:54_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E7:54_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E36:54_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E11:54_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="pencil_1.html#E17:54_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E37:54_1"/>
 <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="E38:54_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</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>16</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E36:54_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E39:54_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E11:54_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="pencil_1.html#E17:54_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E40:54_1"/>
 <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="E41:54_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</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>16</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E39:54_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E42:54_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>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E27:54_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="pencil_1.html#E38:54_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E43:54_1"/><b>then </b>
not <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E44:54_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_1.html#E14:54_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<i><font color="Green">E58</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> = 
<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 href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E46:54_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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>5</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>7</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="54_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_1_6"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>17</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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</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="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_6_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_6_1_1"><b>suppose </b></a><a NAME="E1:54_1_6_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> as   <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>18</sub></font></span>)</span> <b>by </b><i><a class="txt" href="pencil_1.html#E6:54_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="pencil_1.html#E35:54_1"><i><font color="Green">E52</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> as   <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>18</sub></font></span>)</span> <b>by </b><i><a class="txt" href="pencil_1.html#E13:54_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="pencil_1.html#E44:54_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="pencil_1.html#E1:54_1_6_1_1"><i><font color="Green">E60</font></i></a></i>;<br/><a NAME="E4:54_1_6_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>13</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>19</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E30:54_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="pencil_1.html#E38:54_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="pencil_1.html#E1:54_1_6_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:54_1_6_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="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E32:54_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="pencil_1.html#E41:54_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="pencil_1.html#E1:54_1_6_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:54_1_6_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
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>19</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_1.html#E27:54_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="pencil_1.html#E1:54_1_6_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="pencil_1.html#E4:54_1_6_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E7:54_1_6_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> is <a href="pencil_1.html#V6" target="_self">identifying_close_blocks</a>
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:54"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E8:54_1_6_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E6:54_1_6_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="pencil_1.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_6_1_2"><b>suppose </b></a><a NAME="E1:54_1_6_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><a NAME="E2:54_1_6_1_2"/><b>then </b>
( not <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="pencil_1.html#E7:54_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="pencil_1.html#E35:54_1"><i><font color="Green">E52</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:54_1_6_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>5</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"><font color="Maroon">c<sub>19</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/><a NAME="E5:54_1_6_1_2"/>
<font color="Maroon">c<sub>13</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_1.html#E18:54_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="pencil_1.html#E1:54_1_6"><i><font color="Green">E59</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E6:54_1_6_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>13</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>19</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E4:54_1_6_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:54_1_6_1_2"/>
<font color="Maroon">c<sub>13</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>7</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_1.html#E36:54_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="pencil_1.html#E45:54_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="pencil_1.html#E1:54_1_6"><i><font color="Green">E59</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E8:54_1_6_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E4:54_1_6_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="pencil_1.html#E6:54_1_6_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E47:54_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E4:54_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pencil_1.html#E11:54_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="pencil_1.html#E45:54_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="card_3.html#T38">CARD_3:38</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<a NAME="E48:54_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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</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="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="54_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_1_7"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</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="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_7_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_7_1_1"><b>suppose </b></a><a NAME="E1:54_1_7_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> as   <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>18</sub></font></span>)</span> <b>by </b><i><a class="txt" href="pencil_1.html#E6:54_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="pencil_1.html#E35:54_1"><i><font color="Green">E52</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> as   <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>18</sub></font></span>)</span> <b>by </b><i><a class="txt" href="pencil_1.html#E13:54_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="pencil_1.html#E44:54_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="pencil_1.html#E1:54_1_7_1_1"><i><font color="Green">E60</font></i></a></i>;<br/><a NAME="E4:54_1_7_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>13</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>19</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E30:54_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="pencil_1.html#E38:54_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="pencil_1.html#E1:54_1_7_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:54_1_7_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="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E32:54_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="pencil_1.html#E41:54_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="pencil_1.html#E1:54_1_7_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:54_1_7_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
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>19</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_1.html#E27:54_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="pencil_1.html#E1:54_1_7_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="pencil_1.html#E4:54_1_7_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="pencil_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E7:54_1_7_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> is <a href="pencil_1.html#V6" target="_self">identifying_close_blocks</a>
 <b>by </b><i><a class="txt" href="pencil_1.html#E1:54"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E8:54_1_7_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</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>17</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E6:54_1_7_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="pencil_1.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_7_1_2"><b>suppose </b></a><a NAME="E1:54_1_7_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><a NAME="E2:54_1_7_1_2"/><b>then </b>
( not <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="pencil_1.html#E14:54_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="pencil_1.html#E44:54_1"><i><font color="Green">E57</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:54_1_7_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>7</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"><font color="Maroon">c<sub>19</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/><a NAME="E5:54_1_7_1_2"/>
<font color="Maroon">c<sub>13</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>7</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_1.html#E36:54_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="pencil_1.html#E1:54_1_7"><i><font color="Green">E59</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E6:54_1_7_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>13</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>19</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E4:54_1_7_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:54_1_7_1_2"/>
<font color="Maroon">c<sub>13</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_1.html#E18:54_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="pencil_1.html#E45:54_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="pencil_1.html#E1:54_1_7"><i><font color="Green">E59</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E8:54_1_7_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</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>17</sub></font>
 <b>by </b><i><a class="txt" href="pencil_1.html#E4:54_1_7_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="pencil_1.html#E6:54_1_7_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E49:54_1"/>
<font color="Maroon">c<sub>4</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_1.html#E4:54_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pencil_1.html#E11:54_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="pencil_1.html#E45:54_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="card_3.html#T38">CARD_3:38</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:54"/>
 <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#V6" target="_self">identifying_close_blocks</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#D7" target="_self">Def7</a></i>;<br/>


</div>
