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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="incproj.html#V6">2-dimensional</a> <a href="incproj.html#V10">Desarguesian</a> <a href="incproj.html#NM2">IncProjSp</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>













<b>assume </b><a NAME="E1:28"/><i><font color="Green">E20</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<a NAME="E2:28"/><i><font color="Green">E21</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/>
<a NAME="E1:28_1"/><i><font color="Green">E22</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E2:28_1"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self">Th25</a></i>;<br/>
<b>thus </b><a NAME="E3:28_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_1"><i><font color="Green">E22</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:28_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E2:28_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:28"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:28_2"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_2_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:28_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_2_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2_1_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E3:28_2_1_1_1_1"/><i><font color="Green">E28</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="projred1.html#E1:28_2_1_1_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E4:28_2_1_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E5:28_2_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E3:28_2_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_2_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:28_2_1_1_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self">Th25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:28_2_1_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E2:28_2_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:28_2_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_2_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_2_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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

<b>thus </b><a NAME="E2:28_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:28_2_1_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2_1_1_2_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E3:28_2_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="projred1.html#E1:28_2_1_1_2_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:28_2_1_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_2_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2_1_1_2_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:28_2_1_1_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="projred1.html#E1:28_2_1_1_2_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:28_2_1_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_2_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_2_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E2:28_2_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</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="28_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:28_2_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:28_2_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="projred1.html#E1:28_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:28_2_1_2"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="projred1.html#E1:28_2_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E1:28_2_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E2:28_2_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E5:28_2_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E2:28_2_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:28_2_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:28_2_1_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E2:28_2_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="projred1.html#E5:28_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="projred1.html#E5:28_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E11:28_2_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="projred1.html#E5:28_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self">Th23</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:28_2_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="projred1.html#E5:28_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E14:28_2_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E5:28_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="projred1.html#E7:28_2_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E9:28_2_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E15:28_2_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E5:28_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="projred1.html#E7:28_2_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E13:28_2_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E16:28_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="projred1.html#E2:28_2_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="projred1.html#E3:28_2_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="projred1.html#E9:28_2_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="projred1.html#E11:28_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="projred1.html#E13:28_2_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="projred1.html#E14:28_2_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:28_2_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> is   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:28"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>hence </b><a NAME="E4:28_2_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_2_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E2:28_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( not <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_2"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:28_3"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:28_3"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:28_3"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="projred1.html#T8" target="_self">Th8</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E9:28_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self">Th23</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E11:28_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self">Th23</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:28_3"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:28_3"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14">INCPROJ:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E17:28_3"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self">Th23</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E19:28_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E20:28_3"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">c<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E21:28_3"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_3_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:28_3_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E11:28_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E3:28_3_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<a NAME="E4:28_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:28_3_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="projred1.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E6:28_3_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="incproj.html#D10">INCPROJ:def 10</a></i>;<br/>
<a NAME="E7:28_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E5:28_3_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E8:28_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="projred1.html#T26" target="_self">Th26</a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:28_3"/><i><font color="Green">E36</font></i>: 
( not <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_3_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:28_3_2_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E3:28_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><b>hence </b><a NAME="E3:28_3_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E7:28_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E13:28_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E21:28_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_2_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:28_3_2_2"/><i><font color="Green">E40</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:28_3_2_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="projred1.html#E19:28_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E5:28_3_2_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E3:28_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_2_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E6:28_3_2_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E3:28_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_2_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E7:28_3_2_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E19:28_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_2_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_2_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E8:28_3_2_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E11:28_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="projred1.html#E13:28_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E20:28_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/><a NAME="E9:28_3_2_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="projred1.html#E7:28_3_2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="projred1.html#T26" target="_self">Th26</a></i>;<br/><b>hence </b><a NAME="E10:28_3_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="projred1.html#T26" target="_self">Th26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:28_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_2"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>take </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E2:28_3_3"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_3_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:28_3_3_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_3_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_3_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E3:28_3_3_1_1_1_1"/><i><font color="Green">E42</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3_1_1_1_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E4:28_3_3_1_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E5:28_3_3_1_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E3:28_3_3_1_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_3_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_3_1_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:28_3_3_1_1_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self">Th25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:28_3_3_1_1_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_3_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:28_3_3_1_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_3_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_3_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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

<b>thus </b><a NAME="E2:28_3_3_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_3_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_3_1_1_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E3:28_3_3_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3_1_1_2_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_3_1_1_2_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_3_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_3_1_1_2_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:28_3_3_1_1_2_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3_1_1_2_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_3_1_1_2_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_3_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_3_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E2:28_3_3_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</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="28_3_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:28_3_3_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:28_3_3_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:28_3_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_3_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_3_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="projred1.html#T21" target="_self">Th21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:28_3_3_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> is   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:28"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_3_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_3_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_3_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( not <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_3"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>take </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E2:28_3_4"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_4_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:28_3_4_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_4_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_4_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E3:28_3_4_1_1_1_1"/><i><font color="Green">E43</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4_1_1_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E4:28_3_4_1_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E5:28_3_4_1_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E3:28_3_4_1_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_4_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_4_1_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:28_3_4_1_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self">Th25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:28_3_4_1_1_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_4_1_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:28_3_4_1_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_4_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_4_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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

<b>thus </b><a NAME="E2:28_3_4_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_4_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_4_1_1_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E3:28_3_4_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4_1_1_2_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_4_1_1_2_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green">E39</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_4_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_4_1_1_2_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:28_3_4_1_1_2_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4_1_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_4_1_1_2_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_4_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_4_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E2:28_3_4_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</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="28_3_4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:28_3_4_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:28_3_4_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:28_3_4_1_2"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_4_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E5:28_3_4_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_4_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self">Th23</a></i>;<br/>
<b>hence </b><a NAME="E6:28_3_4_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="projred1.html#E3:28_3_4_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="projred1.html#T21" target="_self">Th21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:28_3_4_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> is   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:28"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_4_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_4_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_4_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_4"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( not <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_4"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_5"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E2:28_3_5"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_5_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_5_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_5_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E3:28_3_5_1_1_1"/><i><font color="Green">E43</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E4:28_3_5_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E5:28_3_5_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E3:28_3_5_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_5_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_5_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:28_3_5_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self">Th25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:28_3_5_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_5_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:28_3_5_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_5_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_5_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_5_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</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="28_3_5_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:28_3_5_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_5_1_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:28_3_5_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_5_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E2:28_3_5_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E3:28_3_5_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T27" target="_self">Th27</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_5_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E22:28_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="projred1.html#T27" target="_self">Th27</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_5_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_5_1_2_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:28_3_5_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_5_1_2_2_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E2:28_3_5_1_2_2_1"/><i><font color="Green">E47</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:28_3_5_1_2_2_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="projred1.html#E19:28_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:28_3_5_1_2_2_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E3:28_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_5_1_2_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E6:28_3_5_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E3:28_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E17:28_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E7:28_3_5_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E5:28_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E19:28_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="projred1.html#E22:28_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_2_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_5_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_5_1_2_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:28_3_5_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_5_1_2_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E2:28_3_5_1_2_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E7:28_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E11:28_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="projred1.html#E13:28_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E22:28_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_5_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3_5_1_2_4"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/><a NAME="E2:28_3_5_1_2_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_5_1_2_4_1"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E2:28_3_5_1_2_4_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E4:28_3_5_1_2_4_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_5_1_2_4_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self">Th23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E6:28_3_5_1_2_4_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2_4_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="projred1.html#T23" target="_self">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:28_3_5_1_2_4_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="projred1.html#E11:28_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:28_3_5_1_2_4_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E12:28_3_5_1_2_4_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2_4_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="projred1.html#E6:28_3_5_1_2_4_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E14:28_3_5_1_2_4_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_5_1_2_4_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2_4_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E15:28_3_5_1_2_4_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_4_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E16:28_3_5_1_2_4_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="projred1.html#E13:28_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_4_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2_4_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a>, <a class="ref" href="projred1.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E17:28_3_5_1_2_4_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>22</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_3_5_1_2_4_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>

<a NAME="E2:28_3_5_1_2_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="projred1.html#E3:28_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E8:28_3_5_1_2_4_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="projred1.html#E10:28_3_5_1_2_4_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:28_3_5_1_2_4_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E5:28_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E10:28_3_5_1_2_4_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:28_3_5_1_2_4_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_4_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_3_5_1_2_4_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:28_3_5_1_2_4_1_2"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T27" target="_self">Th27</a></i>;<br/>
<b>hence </b><a NAME="E3:28_3_5_1_2_4_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_4_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="projred1.html#T26" target="_self">Th26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:28_3_5_1_2_4_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_2_4_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_3_5_1_2_4_1_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:28_3_5_1_2_4_1_3"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="projred1.html#E11:28_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5"><i><font color="Green">E39</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E3:28_3_5_1_2_4_1_3"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="projred1.html#T27" target="_self">Th27</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_5_1_2_4_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E7:28_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_5_1_2_4_1_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="projred1.html#T26" target="_self">Th26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:28_3_5_1_2_4_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2_4"><i><font color="Green">E46</font></i></a>, <a class="txt" href="projred1.html#E14:28_3_5_1_2_4_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="projred1.html#E15:28_3_5_1_2_4_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="incproj.html#D9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E21:28_3_5_1_2_4_1"/>
( <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="incproj.html#R1">are_mutually_different</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="projred1.html#E3:28_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="projred1.html#E9:28_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="projred1.html#E15:28_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2_4_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="projred1.html#E8:28_3_5_1_2_4_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="projred1.html#E10:28_3_5_1_2_4_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="projred1.html#E12:28_3_5_1_2_4_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="projred1.html#E14:28_3_5_1_2_4_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="projred1.html#E16:28_3_5_1_2_4_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="projred1.html#E17:28_3_5_1_2_4_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="projred1.html#E18:28_3_5_1_2_4_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="projred1.html#E19:28_3_5_1_2_4_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="incproj.html#D5">INCPROJ:def 5</a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E23:28_3_5_1_2_4_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="incproj.html#R4">on</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E20:28_3_5_1_2_4_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="projred1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E24:28_3_5_1_2_4_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>25</sub></font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E23:28_3_5_1_2_4_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="incproj.html#D8">INCPROJ:def 8</a></i>;<br/>
<a NAME="E25:28_3_5_1_2_4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E26:28_3_5_1_2_4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E22:28_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_5_1_2_4_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="projred1.html#E6:28_3_5_1_2_4_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="projred1.html#E24:28_3_5_1_2_4_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_5_1_2_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:28_3_5_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E2:28_3_5_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="projred1.html#E3:28_3_5_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="projred1.html#E4:28_3_5_1_2"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:28_3_5_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:28_3_5_1_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_5_1_3_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E3:28_3_5_1_3_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_3_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_5_1_3_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E22:28_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:28_3_5_1_3"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3_5_1_3_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:28_3_5_1_3_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E22:28_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="projred1.html#E1:28_3_5_1_3_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="projred1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_5_1_3_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:28"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_3_5_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_5_1_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:28_3_5_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> is   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="incsp_1.html#U1">Points</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="projred1.html#E2:28"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>hence </b><a NAME="E5:28_3_5_1"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:28_3_5_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="projred1.html#E2:28_3_5_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="projred1.html#E3:28_3_5_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_3_5"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( not <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E22:28_3"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E26:28_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( not <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E23:28_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="projred1.html#E24:28_3"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:28"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( not <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred1.html#K1" target="_self">IncProj</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="projred1.html#E5:28"><i><font color="Green">E23</font></i></a></i>;<br/>


</div>
