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<b>let </b><font color="Maroon" title="c1">IPP</font> be  <a href="incproj.html#V6" title="INCPROJ:attr.6">2-dimensional</a> <a href="incproj.html#V10" title="INCPROJ:attr.10">Desarguesian</a> <a href="incproj.html#NM2" title="INCPROJ:NM.2">IncProjSp</a>;<br/><b>let </b><font color="Maroon" title="c2">o</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font>;<br/>







<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E8">A1</font></i>: 
( not <font color="Maroon" title="c2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">A</font> &amp; not <font color="Maroon" title="c2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">B</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c5">f</font> =  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">B</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:9_1"/><i><font color="Green" title="E9">A2</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">B</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:9_1"/><i><font color="Green" title="E10">A3</font></i>: 
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font>
 ;<br/><a NAME="E3:9_1"/><i><font color="Green" title="E11">A4</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="projred2.html#K1" title="PROJRED2:func.1">CHAIN</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="projred2.html#K1" title="PROJRED2:func.1">CHAIN</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="txt" href="projred2.html#E1:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="projred2.html#E1:9_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="projred2.html#T5" target="_self" title="PROJRED2:th.5">Th5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">a</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E5:9_1"/><i><font color="Green" title="E12">A5</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c8">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">A</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c9">b</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E7:9_1"/><i><font color="Green" title="E13">A6</font></i>: 
( <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c9">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="txt" href="projred2.html#E3:9_1"><i><font color="Green" title="E11">A4</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">x</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c11">y</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">b</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred2.html#E1:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="projred2.html#E5:9_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="projred2.html#E7:9_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="projred1.html#T22" title="PROJRED1:th.22">PROJRED1:22</a></i>;<br/><a NAME="E9:9_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="projred2.html#E2:9_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="projred2.html#E5:9_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="projred2.html#E7:9_1"><i><font color="Green" title="E13">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E10:9_1"/>
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font>
 <b>by </b><i><a class="txt" href="projred2.html#E1:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="projred2.html#E5:9_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="projred2.html#E7:9_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="projred1.html#T26" title="PROJRED1:th.26">PROJRED1:26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9"/>
 <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">B</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


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