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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:36"/><i><font color="Green">E34</font></i>: 
 <a href="analmetr.html#R1">Gen</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:36"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="geomtrap.html#R2" target="_self">are_DTr_wrt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:36"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="geomtrap.html#E2:36"><i><font color="Green">E35</font></i></a>, <a class="ref" href="geomtrap.html#D3" target="_self">Def3</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>assume </b><a NAME="E4:36"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E5:36"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="geomtrap.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E6:36"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="geomtrap.html#E3:36"><i><font color="Green">E36</font></i></a>, <a class="ref" href="analoaf.html#T21">ANALOAF:21</a></i>;<br/>
<a NAME="E7:36"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="geomtrap.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E8:36"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="analoaf.html#T17">ANALOAF:17</a>, <a class="ref" href="analoaf.html#T18">ANALOAF:18</a></i>;<br/>
<a NAME="E9:36"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:36_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>hence </b><a NAME="E3:36_1_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="geomtrap.html#E6:36"><i><font color="Green">E39</font></i></a>, <a class="txt" href="geomtrap.html#E7:36"><i><font color="Green">E40</font></i></a>, <a class="ref" href="analoaf.html#T20">ANALOAF:20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:36_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="analoaf.html#T18">ANALOAF:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:36"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:36_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>hence </b><a NAME="E3:36_2_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="geomtrap.html#E3:36"><i><font color="Green">E36</font></i></a>, <a class="txt" href="geomtrap.html#E7:36"><i><font color="Green">E40</font></i></a>, <a class="ref" href="analoaf.html#T20">ANALOAF:20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:36_2"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="analoaf.html#R1">//</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="analoaf.html#T18">ANALOAF:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:36"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="geomtrap.html#E7:36"><i><font color="Green">E40</font></i></a>, <a class="ref" href="geomtrap.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E12:36"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="geomtrap.html#E8:36"><i><font color="Green">E41</font></i></a>, <a class="ref" href="geomtrap.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E13:36"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="geomtrap.html#E9:36"><i><font color="Green">E42</font></i></a>, <a class="ref" href="geomtrap.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E14:36"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="geomtrap.html#E10:36"><i><font color="Green">E43</font></i></a>, <a class="ref" href="geomtrap.html#D1" target="_self">Def1</a></i>;<br/>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:36_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="geomtrap.html#E1:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="geomtrap.html#E11:36"><i><font color="Green">E44</font></i></a>, <a class="txt" href="geomtrap.html#E12:36"><i><font color="Green">E45</font></i></a>, <a class="txt" href="geomtrap.html#E13:36"><i><font color="Green">E46</font></i></a>, <a class="txt" href="geomtrap.html#E37"><i><font color="Green">Lemma32</font></i></a></i>;<br/><a NAME="E3:36_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="geomtrap.html#E1:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="geomtrap.html#E3:36"><i><font color="Green">E36</font></i></a>, <a class="txt" href="geomtrap.html#E1:36_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="geomtrap.html#E27"><i><font color="Green">Lemma24</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_3"/>
contradiction
 <b>by </b><i><a class="txt" href="geomtrap.html#E1:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="geomtrap.html#E5:36"><i><font color="Green">E38</font></i></a>, <a class="txt" href="geomtrap.html#E25"><i><font color="Green">Lemma22</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_4"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:36_4"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="geomtrap.html#E1:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="geomtrap.html#E11:36"><i><font color="Green">E44</font></i></a>, <a class="txt" href="geomtrap.html#E12:36"><i><font color="Green">E45</font></i></a>, <a class="txt" href="geomtrap.html#E14:36"><i><font color="Green">E47</font></i></a>, <a class="txt" href="geomtrap.html#E37"><i><font color="Green">Lemma32</font></i></a></i>;<br/><a NAME="E3:36_4"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="geomtrap.html#K1" target="_self">#</a> <font color="Maroon">c<sub>6</sub></font> <a href="analmetr.html#R3">are_Ort_wrt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="geomtrap.html#E1:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="geomtrap.html#E3:36"><i><font color="Green">E36</font></i></a>, <a class="txt" href="geomtrap.html#E1:36_4"><i><font color="Green">E49</font></i></a>, <a class="txt" href="geomtrap.html#E27"><i><font color="Green">Lemma24</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_4"/>
contradiction
 <b>by </b><i><a class="txt" href="geomtrap.html#E1:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="geomtrap.html#E5:36"><i><font color="Green">E38</font></i></a>, <a class="txt" href="geomtrap.html#E25"><i><font color="Green">Lemma22</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:36"/>
contradiction
 <b>by </b><i><a class="txt" href="geomtrap.html#E4:36"><i><font color="Green">E37</font></i></a>, <a class="txt" href="geomtrap.html#E15:36"><i><font color="Green">E48</font></i></a></i>;<br/>


</div>
