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

<b>let </b><font color="Maroon">SAS</font> be   <a href="semi_af1.html#NM1">Semi_Affine_Space</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/>









<b>assume </b><a NAME="E1:71"/><i><font color="Green">E29</font></i>: 
 <a href="semi_af1.html#R3" target="_self">congr</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 ;<br/>

<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:71_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T33" target="_self">Th33</a>, <a class="ref" href="semi_af1.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E3:71_1"/>
 <a href="semi_af1.html#R3" target="_self">congr</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T28" target="_self">Th28</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/><a NAME="E2:71_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="semi_af1.html#R3" target="_self">congr</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">d</font> &amp;  <a href="semi_af1.html#R3" target="_self">congr</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T33" target="_self">Th33</a>, <a class="ref" href="semi_af1.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E3:71_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E4:71_2"/>
 <a href="semi_af1.html#R3" target="_self">congr</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_3"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  )
 ;<br/><b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E3:71_3"/><i><font color="Green">E89</font></i>: 
(  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T33" target="_self">Th33</a>, </i>;<br/><a NAME="E4:71_3"/>
( not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <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="ref" href="semi_af1.html#T40" target="_self">Th40</a>, <a class="ref" href="semi_af1.html#T41" target="_self">Th41</a>, , , <a class="txt" href="semi_af1.html#E2:71"><i><font color="Green">E30</font></i></a>, <a class="txt" href="semi_af1.html#E4:71"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E5:71_3"/><b>then </b>
not <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">p</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font><b> such that </b><br/><a NAME="E7:71_3"/><i><font color="Green">E90</font></i>: 
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E8:71_3"/><i><font color="Green">E91</font></i>: 
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T42" target="_self">Th42</a>, </i>;<br/><a NAME="E9:71_3"/>
<font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i><a class="ref" href="semi_af1.html#T40" target="_self">Th40</a>, </i>;<br/><a NAME="E10:71_3"/><b>then </b><i><font color="Green">E92</font></i>: 
not <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">b</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i><a class="ref" href="semi_af1.html#T42" target="_self">Th42</a>, </i>;<br/><a NAME="E11:71_3"/>
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T41" target="_self">Th41</a>, <a class="txt" href="semi_af1.html#E1:71"><i><font color="Green">E29</font></i></a>, <a class="txt" href="semi_af1.html#E2:71"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E12:71_3"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i/>;<br/><a NAME="E13:71_3"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T40" target="_self">Th40</a>, <a class="ref" href="semi_af1.html#T42" target="_self">Th42</a>, <a class="txt" href="semi_af1.html#E1:71"><i><font color="Green">E29</font></i></a>, </i>;<br/><a NAME="E14:71_3"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">d</font>
 <b>by </b><i/>;<br/><a NAME="E15:71_3"/>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T41" target="_self">Th41</a>, <a class="txt" href="semi_af1.html#E1:71"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E16:71_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T40" target="_self">Th40</a>, , </i>;<br/><a NAME="E17:71_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T40" target="_self">Th40</a>, <a class="ref" href="semi_af1.html#T42" target="_self">Th42</a>, , <a class="txt" href="semi_af1.html#E1:71"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E18:71_3"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E19:71_3"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">r</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">p</font>
 <b>by </b><i/>;<br/><a NAME="E20:71_3"/>
<font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T40" target="_self">Th40</a>, <a class="ref" href="semi_af1.html#T49" target="_self">Th49</a>, </i>;<br/><a NAME="E21:71_3"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">d</font>
 <b>by </b><i/>;<br/><a NAME="E22:71_3"/>
( <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T41" target="_self">Th41</a>, <a class="ref" href="semi_af1.html#T43" target="_self">Th43</a>, <a class="txt" href="semi_af1.html#E1:71"><i><font color="Green">E29</font></i></a>, <a class="txt" href="semi_af1.html#E2:71"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E23:71_3"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">r</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E24:71_3"/>
<font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T41" target="_self">Th41</a>, <a class="txt" href="semi_af1.html#E1:71"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E25:71_3"/><b>then </b>
<font color="Maroon">q</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">c</font>
 <b>by </b><i/>;<br/><a NAME="E26:71_3"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">r</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T51" target="_self">Th51</a>, <a class="ref" href="semi_af1.html#D3" target="_self">Def3</a>, <a class="ref" href="semi_af1.html#T54" target="_self">Th54</a>, </i>;<br/><a NAME="E27:71_3"/><b>then </b>
(  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T42" target="_self">Th42</a>, <a class="ref" href="semi_af1.html#T44" target="_self">Th44</a>, <a class="ref" href="semi_af1.html#T50" target="_self">Th50</a>, <a class="txt" href="semi_af1.html#E1:71"><i><font color="Green">E29</font></i></a>, </i>;<br/><b>hence </b><a NAME="E28:71_3"/>
 <a href="semi_af1.html#R3" target="_self">congr</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_4"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; not <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  )
 ;<br/><a NAME="E2:71_4"/><b>then </b>
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T33" target="_self">Th33</a>, <a class="ref" href="semi_af1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E3:71_4"/><b>then </b>
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:71_4"/>
 <a href="semi_af1.html#R3" target="_self">congr</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:71"/>
 <a href="semi_af1.html#R3" target="_self">congr</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#D2" target="_self">Def2</a>, <a class="ref" href="semi_af1.html#T37" target="_self">Th37</a>, <a class="ref" href="semi_af1.html#T39" target="_self">Th39</a></i>;<br/>


</div>
