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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="parsp_2.html#NM1">FanodesSp</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>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

















<b>assume </b><a NAME="E1:57"/><i><font color="Green">E48</font></i>: 
(  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <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> &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 ;<br/>

<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_1"/>
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="parsp_2.html#R1" target="_self">is_collinear</a> 
 ;<br/><a NAME="E2:57_1"/><b>then </b>
(  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="parsp_2.html#T48" target="_self">Th48</a></i>;<br/><b>hence </b><a NAME="E3:57_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="parsp_2.html#T47" target="_self">Th47</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_2"/>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="parsp_2.html#R1" target="_self">is_collinear</a> 
 ;<br/><a NAME="E2:57_2"/><b>then </b>
( not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="parsp_2.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E3:57_2"/><b>then </b>
(  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="parsp_2.html#T48" target="_self">Th48</a></i>;<br/><b>hence </b><a NAME="E4:57_2"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="parsp_2.html#T47" target="_self">Th47</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  )
 ;<br/><a NAME="E2:57_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="parsp_2.html#T34" target="_self">Th34</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:57_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="parsp_2.html#T46" target="_self">Th46</a></i>;<br/><a NAME="E5:57_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E4:57_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="parsp_2.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E6:57_3"/><i><font color="Green">E54</font></i>: 
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="parsp_2.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="parsp_2.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E7:57_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="parsp_1.html#T42">PARSP_1:42</a></i>;<br/><a NAME="E8:57_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="parsp_2.html#T34" target="_self">Th34</a></i>;<br/><a NAME="E9:57_3"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="parsp_2.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i><a class="txt" href="parsp_2.html#E4:57_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="parsp_2.html#E5:57_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="parsp_2.html#E6:57_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="parsp_2.html#E7:57_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="parsp_2.html#T17" target="_self">Th17</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:57_3"/><i><font color="Green">E56</font></i>: 
 <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="parsp_2.html#T42" target="_self">Th42</a></i>;<br/><a NAME="E12:57_3"/><i><font color="Green">E57</font></i>: 
 <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57"><i><font color="Green">E48</font></i></a>, <a class="txt" href="parsp_2.html#E4:57_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="parsp_2.html#E11:57_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="parsp_2.html#T49" target="_self">Th49</a></i>;<br/><a NAME="E13:57_3"/><b>then </b>
( not <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  &amp; not <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="parsp_2.html#E11:57_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="parsp_2.html#T37" target="_self">Th37</a></i>;<br/><a NAME="E14:57_3"/><b>then </b>
(  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57"><i><font color="Green">E48</font></i></a>, <a class="txt" href="parsp_2.html#E11:57_3"><i><font color="Green">E56</font></i></a>, <a class="txt" href="parsp_2.html#E12:57_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="parsp_2.html#T48" target="_self">Th48</a></i>;<br/><b>hence </b><a NAME="E15:57_3"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="parsp_2.html#T47" target="_self">Th47</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:57"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E2:57"><i><font color="Green">E49</font></i></a>, <a class="txt" href="parsp_2.html#E3:57"><i><font color="Green">E50</font></i></a></i>;<br/>


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