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<span class="kw">let </span><font color="Maroon" title="c1">FdSp</font> be   <a href="parsp_2.html#NM1" title="PARSP_2:NM.1">FanodesSp</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font>, <font color="Olive" title="b5">r</font>, <font color="Olive" title="b6">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FdSp</font>  st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a>  &amp;  <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Olive" title="b5">r</font>,<font color="Olive" title="b6">s</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds <br/> <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Olive" title="b5">r</font>,<font color="Olive" title="b6">s</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font>, <font color="Maroon" title="c6">r</font>, <font color="Maroon" title="c7">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FdSp</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a>  &amp;  <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> implies  <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:68"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">and </span><br/><a NAME="E2:68"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E3:68"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
 <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:68_1"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FdSp</font><span class="kw"> such that </span><br/><a NAME="E3:68_1"/><span class="lab"><font color="Green" title="E60">A5</font></span>: 
 <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">and </span><br/><a NAME="E4:68_1"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
 <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="parsp_2.html#D4" target="_self" title="PARSP_2:def.4">Def4</a></span>;<br/><a NAME="E5:68_1"/>
( <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="parsp_2.html#D3" target="_self" title="PARSP_2:def.3">Def3</a>, <a class="ref" href="parsp_2.html#T48" target="_self" title="PARSP_2:th.48">Th58</a></span>;<br/><a NAME="E6:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A7</font></span>: 
<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="parsp_1.html#T26" title="PARSP_1:th.26">PARSP_1:26</a></span>;<br/><a NAME="E7:68_1"/><span class="lab"><font color="Green" title="E63">A8</font></span>: 
 <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="parsp_2.html#T33" target="_self" title="PARSP_2:th.33">Th41</a></span>;<br/><a NAME="E8:68_1"/>
 <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="ref" href="parsp_2.html#T33" target="_self" title="PARSP_2:th.33">Th41</a></span>;<br/><a NAME="E9:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A9</font></span>: 
<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c4">c</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_1"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="txt" href="#E7:68_1"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="ref" href="parsp_2.html#T42" target="_self" title="PARSP_2:th.42">Th50</a></span>;<br/><a NAME="E10:68_1"/>
 not <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c4">c</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="parsp_2.html#T29" target="_self" title="PARSP_2:th.29">Th37</a></span>;<br/><span class="kw">hence </span><a NAME="E11:68_1"/>
 <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68_1"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="txt" href="#E9:68_1"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="ref" href="parsp_2.html#D3" target="_self" title="PARSP_2:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:68"/>
 <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="parsp_2.html#T26" target="_self" title="PARSP_2:th.26">Th34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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