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<b>let </b><font color="Maroon">IPP</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">o</font> be   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="incsp_1.html#NM2">LINE</a> of <font color="Maroon">IPP</font>;<br/>







<b>assume </b><a NAME="E1:7"/><i><font color="Green">E47</font></i>: 
( not <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">o</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">B</font> )
 ;<br/>

<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</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">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">o</font>,<font color="Maroon">B</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> ;<br/><a NAME="E3:7_1"/>
<font color="Maroon">x'</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="projred1.html#D1">PROJRED1:def 1</a></i>;<br/><a NAME="E4:7_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> : <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E5:7_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="projred2.html#K1" target="_self">CHAIN</a> <font color="Maroon">A</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:7_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="projred2.html#K1" target="_self">CHAIN</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:7_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> : <font color="Olive">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> </span>}</span> 
 ;<br/><a NAME="E3:7_2"/><b>then </b>
 ex <font color="Olive">b</font> being  <a href="incsp_1.html#NM1">POINT</a> of <font color="Maroon">IPP</font> st <br/>( <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> )
 ;<br/><b>hence </b><a NAME="E4:7_2"/>
<font color="Maroon">x</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">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">o</font>,<font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="projred1.html#D1">PROJRED1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:7"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1">IncProj</a> <font color="Maroon">A</font>,<font color="Maroon">o</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="projred2.html#K1" target="_self">CHAIN</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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