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<b>let </b><font color="Maroon">FdSp</font> be   <a href="parsp_2.html#NM1">FanodesSp</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FdSp</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FdSp</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FdSp</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FdSp</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FdSp</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FdSp</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FdSp</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FdSp</font>;<br/>

















<b>assume </b><a NAME="E1:57"/><i><font color="Green">E17</font></i>: 
(  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">q</font> &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">r</font> &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">d</font>,<font color="Maroon">s</font> )
 ;<br/>

<div><i><font color="Green">E18</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">b</font>,<font color="Maroon">q</font>,<font color="Maroon">c</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">b</font>,<font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">r</font> &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">d</font>,<font color="Maroon">s</font> )
 <b>by </b><i>, <a class="txt" href="parsp_2.html#E3:57"><i><font color="Green">E45</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_1"/>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E4"><i><font color="Green">Lemma44</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E45</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">a</font>,<font color="Maroon">p</font>,<font color="Maroon">d</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">a</font>,<font color="Maroon">p</font>,<font color="Maroon">d</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">b</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font> &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">d</font>,<font color="Maroon">s</font> &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">r</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:57_2"/><b>then </b>
(  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">d</font>,<font color="Maroon">s</font> &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">r</font> )
 <b>by </b><i><a class="txt" href="parsp_2.html#E3:57"><i><font color="Green">E45</font></i></a></i>;<br/><b>hence </b><a NAME="E4:57_2"/>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E4"><i><font color="Green">Lemma44</font></i></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">E46</font></i>: 
( <font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">c</font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">d</font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  )
 ;<br/><a NAME="E2:57_3"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E5:57_3"><i><font color="Green">E48</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FdSp</font><b> such that </b><br/><a NAME="E4:57_3"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><a NAME="E5:57_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E12:57_3"><i><font color="Green">E52</font></i></a>, </i>;<br/><a NAME="E6:57_3"/><i><font color="Green">E49</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="parsp_2.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i><a class="txt" href="parsp_2.html#E5:57_3"><i><font color="Green">E48</font></i></a>, </i>;<br/><a NAME="E7:57_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">a</font>,<font color="Maroon">p</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">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E5:57_3"><i><font color="Green">E48</font></i></a>, </i>;<br/><a NAME="E9:57_3"/><b>then </b>
not <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="parsp_2.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i><a class="txt" href="parsp_2.html#E12:57_3"><i><font color="Green">E52</font></i></a>, , , , </i>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FdSp</font><b> such that </b><br/><a NAME="E11:57_3"/><i><font color="Green">E51</font></i>: 
 <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i/>;<br/><a NAME="E12:57_3"/><i><font color="Green">E52</font></i>: 
 <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">b</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E5:57_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="parsp_2.html#E12:57_3"><i><font color="Green">E52</font></i></a>, , <a class="txt" href="parsp_2.html#E1:57_3"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E13:57_3"/><b>then </b>
( not <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  &amp; not <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">d</font> <a href="parsp_2.html#R1" target="_self">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="parsp_2.html#E11:57_3"><i><font color="Green">E51</font></i></a>, , </i>;<br/><a NAME="E14:57_3"/><b>then </b>
(  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<font color="Maroon">r</font> &amp;  <a href="parsp_2.html#R2" target="_self">parallelogram</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">d</font>,<font color="Maroon">s</font> )
 <b>by </b><i><a class="txt" href="parsp_2.html#E5:57_3"><i><font color="Green">E48</font></i></a>, , , <a class="txt" href="parsp_2.html#E3:57"><i><font color="Green">E45</font></i></a></i>;<br/><b>hence </b><a NAME="E15:57_3"/>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E4"><i><font color="Green">Lemma44</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:57"/>
<font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="parsp_1.html#R1">'||'</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/>


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