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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</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">K</font>, <font color="Olive" title="b3">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><br/> for <font color="Olive" title="b4">a1</font>, <font color="Olive" title="b5">b1</font>, <font color="Olive" title="b6">a2</font>, <font color="Olive" title="b7">b2</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>  st not <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">K</font> &amp; not <font color="Olive" title="b1">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">L</font> &amp; <font color="Olive" title="b4">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">K</font> &amp; <font color="Olive" title="b5">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">K</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Olive" title="b2">K</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">a2</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Olive" title="b2">K</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">b2</font> &amp; <font color="Olive" title="b6">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">b2</font> holds <br/><font color="Olive" title="b4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">b1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">K</font>, <font color="Olive" title="b2">L</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><br/> for <font color="Olive" title="b3">a1</font>, <font color="Olive" title="b4">b1</font>, <font color="Olive" title="b5">a2</font>, <font color="Olive" title="b6">b2</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>  st not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">L</font> &amp; <font color="Olive" title="b3">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b4">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">K</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b2">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">a2</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b2">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">b2</font> &amp; <font color="Olive" title="b5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">b2</font> holds <br/><font color="Olive" title="b3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font>, <font color="Maroon" title="c4">L</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">a2</font>, <font color="Olive" title="b4">b2</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>  st not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Olive" title="b1">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Olive" title="b2">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a2</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b2</font> &amp; <font color="Olive" title="b3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b2</font> holds <br/><font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">a1</font>, <font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c7">a2</font>, <font color="Maroon" title="c8">b2</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="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c5">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a2</font> &amp; <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b2</font> &amp; <font color="Maroon" title="c7">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b2</font> implies <font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:29"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">and </span><br/><a NAME="E2:29"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
not <font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L</font>
 <span class="kw">and </span><br/><a NAME="E3:29"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c5">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">and </span><br/><a NAME="E4:29"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c6">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">and </span><br/><a NAME="E5:29"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a2</font>
 <span class="kw">and </span><br/><a NAME="E6:29"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b2</font>
 <span class="kw">and </span><br/><a NAME="E7:29"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<font color="Maroon" title="c7">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">a2</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a1</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="#E5:29"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>;<br/>
<a NAME="E9:29"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c9">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">A</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E11:29"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">A</font>
 <span class="kw">and </span><br/><a NAME="E12:29"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c5">a1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">A</font>
 <span class="kw">and </span><br/><a NAME="E13:29"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<font color="Maroon" title="c9">a2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">b2</font> = <span class="p1">(<span class="default"><a href="projred1.html#K1" title="PROJRED1:func.1">IncProj</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>;<br/>
<a NAME="E15:29"/>
<font color="Maroon" title="c11">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="projred1.html#T23" target="_self" title="PROJRED1:th.23">Th23</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">B</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><span class="kw"> such that </span><br/><a NAME="E17:29"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">B</font>
 <span class="kw">and </span><br/><a NAME="E18:29"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<font color="Maroon" title="c6">b1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">B</font>
 <span class="kw">and </span><br/><a NAME="E19:29"/><span class="lab"><font color="Green" title="E32">A14</font></span>: 
<font color="Maroon" title="c11">b2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="projred1.html#D1" target="_self" title="PROJRED1:def.1">Def1</a></span>;<br/>
<a NAME="E20:29"/>
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E9:29"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E11:29"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E13:29"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E17:29"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E19:29"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:29"/>
<font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E11:29"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E12:29"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E18:29"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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