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

<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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">q</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">r</font>, <font color="Olive" title="b7">d</font>, <font color="Olive" title="b8">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  <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">q</font> &amp;  <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">p</font>,<font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">r</font> &amp;  <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">q</font>,<font color="Olive" title="b7">d</font>,<font color="Olive" title="b8">s</font> holds <br/><font color="Olive" title="b5">c</font>,<font color="Olive" title="b7">d</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Olive" title="b6">r</font>,<font color="Olive" title="b8">s</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">r</font>, <font color="Maroon" title="c8">d</font>, <font color="Maroon" title="c9">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="Red">::  thesis: </font></span></a><span class="hide"> (  <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">p</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</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">p</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">r</font> &amp;  <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">s</font> implies <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c9">s</font> )</span><br/>

















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:65"/><span class="lab"><font color="Green" title="E48">A1</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">p</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>
 <span class="kw">and </span><br/><a NAME="E2:65"/><span class="lab"><font color="Green" title="E49">A2</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">p</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">r</font>
 <span class="kw">and </span><br/><a NAME="E3:65"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
 <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c9">s</font></span><br/>

<div><span class="lab"><font color="Green" title="E51">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:65_1"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">d</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c9">s</font></span><br/><a NAME="E2:65_1"/>
 <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="parsp_2.html#T41" target="_self" title="PARSP_2:th.41">Th41</a></span>;<br/><a NAME="E3:65_1"/><span class="kw">then </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">p</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="ref" href="parsp_2.html#T48" target="_self" title="PARSP_2:th.48">Th48</a></span>;<br/><span class="kw">hence </span><a NAME="E4:65_1"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c9">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="parsp_2.html#T47" target="_self" title="PARSP_2:th.47">Th47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E52">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:65_2"/><span class="lab"><font color="Green" title="E53">A7</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">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="parsp_2.html#T34" target="_self" title="PARSP_2:th.34">Th34</a></span>;<br/><a NAME="E2:65_2"/><span class="lab"><font color="Green" title="E54">A8</font></span>: 
( not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">b</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">p</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">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="parsp_2.html#T36" target="_self" title="PARSP_2:th.36">Th36</a>, <a class="ref" href="parsp_1.html#T42" title="PARSP_1:th.42">PARSP_1:42</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:65_2"/><span class="lab"><font color="Green" title="E55">A9</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">c</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E4:65_2"/><span class="lab"><font color="Green" title="E56">A10</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">d</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c9">s</font></span><br/><a NAME="E5:65_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="parsp_2.html#T34" target="_self" title="PARSP_2:th.34">Th34</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</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="E7:65_2"/><span class="lab"><font color="Green" title="E57">A11</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c10">x</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E8:65_2"/><span class="lab"><font color="Green" title="E58">A12</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E9:65_2"/><span class="lab"><font color="Green" title="E59">A13</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="parsp_2.html#T46" target="_self" title="PARSP_2:th.46">Th46</a></span>;<br/><a NAME="E10:65_2"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:65_2"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="ref" href="parsp_2.html#D2" target="_self" title="PARSP_2:def.2">Def2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">y</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="E12:65_2"/><span class="lab"><font color="Green" title="E60">A14</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">p</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:65_2"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="txt" href="#E2:65_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E1:65_2"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="ref" href="parsp_2.html#T17" target="_self" title="PARSP_2:th.17">Th17</a>, <a class="ref" href="parsp_2.html#T42" target="_self" title="PARSP_2:th.42">Th42</a></span>;<br/><a NAME="E13:65_2"/><span class="lab"><font color="Green" title="E61">A15</font></span>: 
not <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c8">d</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="#E4:65_2"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E12:65_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="parsp_2.html#T37" target="_self" title="PARSP_2:th.37">Th37</a></span>;<br/><a NAME="E14:65_2"/>
 <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E7:65_2"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="txt" href="#E9:65_2"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E12:65_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="parsp_2.html#T49" target="_self" title="PARSP_2:th.49">Th49</a></span>;<br/><a NAME="E15:65_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A16</font></span>: 
 <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c9">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E13:65_2"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="ref" href="parsp_2.html#T48" target="_self" title="PARSP_2:th.48">Th48</a></span>;<br/><a NAME="E16:65_2"/>
not <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c6">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="#E1:65"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E3:65_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E7:65_2"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="txt" href="#E9:65_2"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E12:65_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="parsp_2.html#T37" target="_self" title="PARSP_2:th.37">Th37</a>, <a class="ref" href="parsp_2.html#T49" target="_self" title="PARSP_2:th.49">Th49</a></span>;<br/><a NAME="E17:65_2"/><span class="kw">then </span>
 <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E12:65_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="parsp_2.html#T48" target="_self" title="PARSP_2:th.48">Th48</a></span>;<br/><span class="kw">hence </span><a NAME="E18:65_2"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c9">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:65_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="ref" href="parsp_2.html#T47" target="_self" title="PARSP_2:th.47">Th47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:65_3"/>
not <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">c</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c9">s</font></span><br/><a NAME="E2:65_3"/><span class="kw">then </span>
 <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="parsp_2.html#T48" target="_self" title="PARSP_2:th.48">Th48</a></span>;<br/><span class="kw">hence </span><a NAME="E3:65_3"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c9">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="parsp_2.html#T47" target="_self" title="PARSP_2:th.47">Th47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:65"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">d</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c9">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E5:65"><span class="lab"><font color="Green" title="E52">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
