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<span class="kw">let </span><font color="Maroon" title="c1">IPP</font> be   <a href="incproj.html#V5" title="INCPROJ:attr.5">2-dimensional</a>   <a href="incproj.html#V9" title="INCPROJ:attr.9">Desarguesian</a>  <a href="incproj.html#NM2" title="INCPROJ:NM.2">IncProjSp</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font><br/>  for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font>  st  not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">A</font> &amp;  not <font color="Olive" title="b1">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">B</font> holds <br/> <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> (<font color="Olive" title="b2">A</font>,<font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">B</font>) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font>  st  not <font color="Maroon" title="c2">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">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="Olive" title="b2">B</font> holds <br/> <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> (<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c2">o</font>,<font color="Olive" title="b2">B</font>) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  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> implies  <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>  )</span><br/>







<span class="kw">set </span><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/>
<span class="kw">assume </span><a NAME="E1:9"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
(  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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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; <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> implies <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:9_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E2:9_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E3:9_1"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font></span><br/><a NAME="E4:9_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E1:9_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="projred2.html#T4" target="_self" title="PROJRED2:th.4">Th5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:9_1"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">and </span><br/><a NAME="E7:9_1"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<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/><a NAME="E8:9_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="projred2.html#T4" target="_self" title="PROJRED2:th.4">Th5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E10:9_1"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">and </span><br/><a NAME="E11:9_1"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<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>
 ;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E7:9_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="#E11:9_1"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="projred1.html#T19" title="PROJRED1:th.19">PROJRED1:19</a></span>;<br/><a NAME="E13:9_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E6:9_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E10:9_1"><span class="lab"><font color="Green" title="E14">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:9_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E6:9_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E7:9_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="#E10:9_1"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="txt" href="#E11:9_1"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="projred1.html#T23" title="PROJRED1:th.23">PROJRED1:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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