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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>













<b>assume </b><a NAME="E1:68"/><i><font color="Green" title="E56">A1</font></i>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_1"/><i><font color="Green" title="E57">A2</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/><b>then consider </b><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><b> such that </b><br/><a NAME="E3:68_1"/><i><font color="Green" title="E58">A3</font></i>: 
(  <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> &amp;  <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> )
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="parsp_2.html#D4" target="_self" title="PARSP_2:def.4">Def4</a></i>;<br/><a NAME="E4:68_1"/><i><font color="Green" title="E59">A4</font></i>: 
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> 
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="parsp_2.html#T37" target="_self" title="PARSP_2:th.37">Th37</a></i>;<br/><a NAME="E5:68_1"/>
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="#E1:68_1"><i><font color="Green" title="E57">A2</font></i></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#T58" target="_self" title="PARSP_2:th.58">Th58</a></i>;<br/><a NAME="E6:68_1"/><b>then </b><i><font color="Green" title="E60">A5</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="parsp_1.html#T43" title="PARSP_1:th.43">PARSP_1:43</a></i>;<br/><a NAME="E7:68_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="c6">r</font>,<font color="Maroon" title="c7">s</font> &amp;  <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> &amp;  <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> )
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="#E3:68_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="parsp_2.html#T41" target="_self" title="PARSP_2:th.41">Th41</a></i>;<br/><a NAME="E8:68_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="parsp_2.html#T50" target="_self" title="PARSP_2:th.50">Th50</a></i>;<br/><b>hence </b><a NAME="E9:68_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>
 <b>by </b><i><a class="txt" href="#E4:68_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="#E6:68_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="parsp_2.html#D3" target="_self" title="PARSP_2:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:68"/>
 <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>
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="parsp_2.html#T34" target="_self" title="PARSP_2:th.34">Th34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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