<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:39"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 ;<br/>

<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:39_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="incsp_1.html#U3">Inc</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="incsp_1.html#D1">INCSP_1:def 1</a></i>;<br/><a NAME="E3:39_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K11" target="_self">Proj_Inc</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:39"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E4:39_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  or  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,1</span><a href="tarski.html#K4">]</a></span> or ( not <!--HUMANRECFAILED-->(  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; not  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:39_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp;  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><a NAME="E2:39_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp;  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:39_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E3:39_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K1" target="_self">AfLines</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K2" target="_self">AfPlanes</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:39_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="afproj.html#T7" target="_self">Th7</a>, <a class="ref" href="afproj.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E4:39_1_1"/><i><font color="Green">E39</font></i>: 
(  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:39"><i><font color="Green">E32</font></i></a>, <a class="txt" href="afproj.html#E1:39_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="afproj.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E5:39_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:39_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="afproj.html#E3:39_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:39_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:39_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="afproj.html#E4:39_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="afproj.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E8:39_1_1"/><i><font color="Green">E41</font></i>: 
(  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K4" target="_self">PlanesParallelity</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K4" target="_self">PlanesParallelity</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:39"><i><font color="Green">E32</font></i></a>, <a class="txt" href="afproj.html#E1:39_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="afproj.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E9:39_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K4" target="_self">PlanesParallelity</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E2:39_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="afproj.html#E3:39_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/><a NAME="E10:39_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:39_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="afproj.html#E8:39_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="afproj.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E11:39_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:39_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="afproj.html#E7:39_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/><a NAME="E12:39_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:39_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_4.html#T56">AFF_4:56</a></i>;<br/><a NAME="E13:39_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E1:39_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_4.html#T59">AFF_4:59</a></i>;<br/><b>hence </b><a NAME="E14:39_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E7:39_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="afproj.html#E10:39_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="afproj.html#E12:39_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="aff_4.html#T60">AFF_4:60</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:39_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E3:39_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="afproj.html#E4:39_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="afproj.html#D11" target="_self">Def11</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:39_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K11" target="_self">Proj_Inc</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:39"><i><font color="Green">E32</font></i></a>, <a class="ref" href="afproj.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E3:39_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="incsp_1.html#U3">Inc</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="afproj.html#E1:39"><i><font color="Green">E32</font></i></a></i>;<br/><b>hence </b><a NAME="E4:39_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D1">INCSP_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E2:39"><i><font color="Green">E33</font></i></a></i>;<br/>


</div>
