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<b>let </b><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>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">o</font>, <font color="Maroon" title="c7">o''</font>, <font color="Maroon" title="c8">d</font>, <font color="Maroon" title="c9">o'</font>, <font color="Maroon" title="c10">oo'</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="c11">A</font>, <font color="Maroon" title="c12">C</font>, <font color="Maroon" title="c13">B</font>, <font color="Maroon" title="c14">Q</font>, <font color="Maroon" title="c15">O</font>, <font color="Maroon" title="c16">O1</font>, <font color="Maroon" title="c17">O2</font>, <font color="Maroon" title="c18">O3</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="E1:17"/><i><font color="Green" title="E12">A1</font></i>: 
( not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> &amp; not <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c13">B</font>,<font color="Maroon" title="c12">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">Q</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">o</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c11">A</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c7">o''</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c13">B</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">o'</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c12">C</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">d</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c15">O</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c10">oo'</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">Q</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c9">o'</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">O1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c9">o'</font>,<font color="Maroon" title="c10">oo'</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c17">O2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c10">oo'</font>,<font color="Maroon" title="c4">q</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c18">O3</font> &amp; <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font> )
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
( not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> &amp; not <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c13">B</font>,<font color="Maroon" title="c12">C</font> <a href="projred2.html#R1" title="PROJRED2:pred.1">are_concurrent</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon" title="c5">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font> &amp; <font color="Maroon" title="c7">o''</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font> &amp; <font color="Maroon" title="c8">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font> &amp; <font color="Maroon" title="c5">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; <font color="Maroon" title="c8">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; <font color="Maroon" title="c9">o'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font> &amp; <font color="Maroon" title="c8">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font> &amp; <font color="Maroon" title="c5">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon" title="c10">oo'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">O1</font> &amp; <font color="Maroon" title="c6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">O1</font> &amp; <font color="Maroon" title="c9">o'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c16">O1</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">O2</font> &amp; <font color="Maroon" title="c9">o'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">O2</font> &amp; <font color="Maroon" title="c10">oo'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">O2</font> &amp; <font color="Maroon" title="c6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> &amp; <font color="Maroon" title="c10">oo'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> &amp; <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> &amp; <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font> )
 
<b>by </b><i><a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/>
<a NAME="E3:17"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c5">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c5">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; <font color="Maroon" title="c5">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:17"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/>
<a NAME="E4:17"/><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c8">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font> &amp; <font color="Maroon" title="c8">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> &amp; <font color="Maroon" title="c8">d</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:17"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/>
<a NAME="E5:17"/><i><font color="Green" title="E16">A5</font></i>: 
( <font color="Maroon" title="c6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:17"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a></i>;<br/>
<a NAME="E6:17"/><i><font color="Green" title="E17">A6</font></i>: 
( <font color="Maroon" title="c10">oo'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon" title="c10">oo'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c17">O2</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:17"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="incsp_1.html#T11" title="INCSP_1:th.11">INCSP_1:11</a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/>
<a NAME="E7:17"/><i><font color="Green" title="E18">A7</font></i>: 
( <font color="Maroon" title="c6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> &amp; <font color="Maroon" title="c10">oo'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:17"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/>
<a NAME="E8:17"/><i><font color="Green" title="E19">A8</font></i>: 
( <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c18">O3</font> &amp; <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:17"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/>
<a NAME="E9:17"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:17"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></i>;<br/>
<a NAME="E10:17"/><b>then </b><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon" title="c9">o'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:17"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E11:17"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">d</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:17"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="projred2.html#D1" target="_self" title="PROJRED2:def.1">Def1</a></i>;<br/>
<a NAME="E12:17"/><i><font color="Green" title="E23">A12</font></i>: 
<font color="Maroon" title="c16">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">O2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1"/>
not <font color="Maroon" title="c16">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">O2</font>
 ;<br/>

<a NAME="E2:17_1"/><b>then </b>
<font color="Maroon" title="c6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c15">O</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:17"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E3:17_1"/><b>then </b>
<font color="Maroon" title="c15">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E4:17"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="projred2.html#E5:17"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred2.html#E11:17"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:17_1"/>
contradiction
 <b>by </b><i><a class="txt" href="projred2.html#E1:17"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:17"/><i><font color="Green" title="E24">A13</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">o</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:17"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred2.html#E11:17"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E14:17"/><i><font color="Green" title="E25">A14</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">oo'</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:17"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred2.html#E10:17"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E15:17"/><i><font color="Green" title="E26">A15</font></i>: 
not <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_2"/>
<font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font>
 ;<br/>

<a NAME="E2:17_2"/><b>then </b>
<font color="Maroon" title="c10">oo'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E5:17"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred2.html#E7:17"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="projred2.html#E8:17"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="projred2.html#E13:17"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:17_2"/>
contradiction
 <b>by </b><i><a class="txt" href="projred2.html#E1:17"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred2.html#E3:17"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projred2.html#E6:17"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="projred2.html#E14:17"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:17"/>
<font color="Maroon" title="c9">o'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_3"/>
not <font color="Maroon" title="c9">o'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">d</font>
 ;<br/>

<a NAME="E2:17_3"/><b>then </b>
<font color="Maroon" title="c16">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">O</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:17"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:17_3"/>
contradiction
 <b>by </b><i><a class="txt" href="projred2.html#E2:17"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred2.html#E11:17"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:17"/><b>then </b>
<font color="Maroon" title="c15">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">O2</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:17"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E18:17"/><b>then </b><i><font color="Green" title="E27">A16</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">oo'</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E2:17"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<a NAME="E19:17"/><i><font color="Green" title="E28">A17</font></i>: 
not <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_4"/>
<font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font>
 ;<br/>

<a NAME="E2:17_4"/><b>then </b>
<font color="Maroon" title="c6">o</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font>
 
<b>by </b><i><a class="txt" href="projred2.html#E6:17"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="projred2.html#E7:17"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="projred2.html#E8:17"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="projred2.html#E18:17"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:17_4"/>
contradiction
 <b>by </b><i><a class="txt" href="projred2.html#E1:17"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred2.html#E3:17"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="projred2.html#E5:17"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="projred2.html#E9:17"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c19">f</font> =  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">C</font>;<br/>
<b>set </b><font color="Maroon" title="c20">g</font> =  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font>;<br/>
<b>set </b><font color="Maroon" title="c21">f1</font> =  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">Q</font>;<br/>
<b>set </b><font color="Maroon" title="c22">g1</font> =  <a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font>;<br/>
<a NAME="E20:17"/><i><font color="Green" title="E29">A18</font></i>: 
(  <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="c11">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred2.html#K1" title="PROJRED2:func.1">CHAIN</a> <font color="Maroon" title="c11">A</font> &amp;  <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="c12">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred2.html#K1" title="PROJRED2:func.1">CHAIN</a> <font color="Maroon" title="c12">C</font> &amp;  <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="c11">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred2.html#K1" title="PROJRED2:func.1">CHAIN</a> <font color="Maroon" title="c11">A</font> &amp;  <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="c14">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred2.html#K1" title="PROJRED2:func.1">CHAIN</a> <font color="Maroon" title="c14">Q</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:17"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred2.html#E15:17"><i><font color="Green" title="E26">A15</font></i></a>, <a class="txt" href="projred2.html#E19:17"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="projred2.html#T5" target="_self" title="PROJRED2:th.5">Th5</a></i>;<br/>
<a NAME="E21:17"/><i><font color="Green" title="E30">A19</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>  st <font color="Olive" title="b1">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x</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>assume </b><a NAME="E1:17_5"/><i><font color="Green" title="E31">A20</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c24">Q1</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E3:17_5"/><i><font color="Green" title="E32">A21</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">Q1</font> &amp; <font color="Maroon" title="c23">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">Q1</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">x'</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="E5:17_5"/><i><font color="Green" title="E33">A22</font></i>: 
( <font color="Maroon" title="c25">x'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c24">Q1</font> &amp; <font color="Maroon" title="c25">x'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">C</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14" title="INCPROJ:def.14">INCPROJ:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">Q2</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E7:17_5"/><i><font color="Green" title="E34">A23</font></i>: 
( <font color="Maroon" title="c25">x'</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c26">Q2</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c26">Q2</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c27">x''</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="E9:17_5"/><i><font color="Green" title="E35">A24</font></i>: 
( <font color="Maroon" title="c27">x''</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c26">Q2</font> &amp; <font color="Maroon" title="c27">x''</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c13">B</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14" title="INCPROJ:def.14">INCPROJ:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">y</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="E11:17_5"/><i><font color="Green" title="E36">A25</font></i>: 
( <font color="Maroon" title="c28">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon" title="c28">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c26">Q2</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D14" title="INCPROJ:def.14">INCPROJ:def 14</a></i>;<br/>
<a NAME="E12:17_5"/>
( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c9">o'</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">oo'</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c17">O2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c9">o'</font>,<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c16">O1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c9">o'</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c25">x'</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c12">C</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c23">x</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c11">A</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c28">y</font>,<font color="Maroon" title="c10">oo'</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c14">Q</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c10">oo'</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c18">O3</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c25">x'</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c24">Q1</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c28">y</font>,<font color="Maroon" title="c25">x'</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c26">Q2</font> &amp; <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c15">O</font> &amp; <font color="Maroon" title="c17">O2</font>,<font color="Maroon" title="c16">O1</font>,<font color="Maroon" title="c12">C</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_different</a>  )
 
<b>by </b><i><a class="txt" href="projred2.html#E2:17"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred2.html#E12:17"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="projred2.html#E1:17_5"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="projred2.html#E3:17_5"><i><font color="Green" title="E32">A21</font></i></a>, <a class="txt" href="projred2.html#E5:17_5"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="projred2.html#E7:17_5"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="projred2.html#E11:17_5"><i><font color="Green" title="E36">A25</font></i></a>, <a class="ref" href="zfmisc_1.html#D5" title="ZFMISC_1:def.5">ZFMISC_1:def 5</a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c29">R</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E14:17_5"/><i><font color="Green" title="E37">A26</font></i>: 
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c28">y</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c23">x</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="incsp_1.html#R4" title="INCSP_1:pred.4">on</a> <font color="Maroon" title="c29">R</font>
 <b>by </b><i><a class="txt" href="projred2.html#E2:17"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="projred2.html#E10:17"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="projred1.html#T14" title="PROJRED1:th.14">PROJRED1:14</a></i>;<br/>
<a NAME="E15:17_5"/><i><font color="Green" title="E38">A27</font></i>: 
( <font color="Maroon" title="c28">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c29">R</font> &amp; <font color="Maroon" title="c4">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c29">R</font> &amp; <font color="Maroon" title="c23">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c29">R</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E14:17_5"><i><font color="Green" title="E37">A26</font></i></a>, <a class="ref" href="incsp_1.html#T12" title="INCSP_1:th.12">INCSP_1:12</a></i>;<br/>
<a NAME="E16:17_5"/><i><font color="Green" title="E39">A28</font></i>: 
( <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x'</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">x''</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:17"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred2.html#E1:17_5"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="projred2.html#E3:17_5"><i><font color="Green" title="E32">A21</font></i></a>, <a class="txt" href="projred2.html#E5:17_5"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="projred2.html#E7:17_5"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="projred2.html#E9:17_5"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="projred1.html#D1" title="PROJRED1:def.1">PROJRED1:def 1</a></i>;<br/>
<a NAME="E17:17_5"/><i><font color="Green" title="E40">A29</font></i>: 
( <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">x''</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:17"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred2.html#E15:17"><i><font color="Green" title="E26">A15</font></i></a>, <a class="txt" href="projred2.html#E19:17"><i><font color="Green" title="E28">A17</font></i></a>, <a class="txt" href="projred2.html#E1:17_5"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="projred2.html#E7:17_5"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="projred2.html#E9:17_5"><i><font color="Green" title="E35">A24</font></i></a>, <a class="txt" href="projred2.html#E11:17_5"><i><font color="Green" title="E36">A25</font></i></a>, <a class="txt" href="projred2.html#E15:17_5"><i><font color="Green" title="E38">A27</font></i></a>, <a class="ref" href="projred1.html#D1" title="PROJRED1:def.1">PROJRED1:def 1</a></i>;<br/>
<a NAME="E18:17_5"/><i><font color="Green" title="E41">A30</font></i>: 
( <font color="Maroon" title="c23">x</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="c11">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">C</font></span>)</span> &amp; <font color="Maroon" title="c25">x'</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="c12">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> &amp; <font color="Maroon" title="c28">y</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="c14">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> &amp; <font color="Maroon" title="c23">x</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="c11">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">Q</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="projred2.html#E20:17"><i><font color="Green" title="E29">A18</font></i></a>, <a class="txt" href="projred2.html#E1:17_5"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="projred2.html#E5:17_5"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="projred2.html#E11:17_5"><i><font color="Green" title="E36">A25</font></i></a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> = 
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font></span>)</span>
<b>by </b><i><a class="txt" href="projred2.html#E16:17_5"><i><font color="Green" title="E39">A28</font></i></a>, <a class="txt" href="projred2.html#E17:17_5"><i><font color="Green" title="E40">A29</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
<b>by </b><i><a class="txt" href="projred2.html#E18:17_5"><i><font color="Green" title="E41">A30</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E20:17_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c23">X</font> =  <a href="projred2.html#K1" title="PROJRED2:func.1">CHAIN</a> <font color="Maroon" title="c11">A</font>;<br/>
<a NAME="E22:17"/><i><font color="Green" title="E31">A31</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred2.html#K1" title="PROJRED2:func.1">CHAIN</a> <font color="Maroon" title="c11">A</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">Q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="projred2.html#K1" title="PROJRED2:func.1">CHAIN</a> <font color="Maroon" title="c11">A</font> )
 
<b>by </b><i><a class="txt" href="projred2.html#E1:17"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="projred2.html#E15:17"><i><font color="Green" title="E26">A15</font></i></a>, <a class="txt" href="projred2.html#E19:17"><i><font color="Green" title="E28">A17</font></i></a>, <a class="txt" href="projred2.html#E20:17"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="projred1.html#T25" title="PROJRED1:th.25">PROJRED1:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c24">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:17_6"/>
<font color="Maroon" title="c24">y</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="c11">A</font>
 ;<br/><a NAME="E2:17_6"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c24">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">A</font> )
 ;<br/><b>hence </b><a NAME="E3:17_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font>
 <b>by </b><i><a class="txt" href="projred2.html#E21:17"><i><font color="Green" title="E30">A19</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E24:17"/>
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">C</font></span>)</span> <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="c14">Q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">B</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="projred2.html#E22:17"><i><font color="Green" title="E31">A31</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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