<?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="firebrick">::  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="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="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="E1:57"/><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="E2:57"/><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="E3:57"/><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="firebrick">::  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="57_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:57_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="firebrick">::  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="E2:57_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="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="parsp_2.html#T33" target="_self" title="PARSP_2:th.33">Th41</a></span>;<br/><a NAME="E3:57_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="#E3:57"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E1:57_1"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="ref" href="parsp_2.html#T40" target="_self" title="PARSP_2:th.40">Th48</a></span>;<br/><span class="kw">hence </span><a NAME="E4:57_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="#E2:57"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="parsp_2.html#T39" target="_self" title="PARSP_2:th.39">Th47</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>
<div><span class="lab"><font color="Green" title="E52">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:57_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="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="parsp_2.html#T26" target="_self" title="PARSP_2:th.26">Th34</a></span>;<br/><a NAME="E2:57_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="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="parsp_2.html#T28" target="_self" title="PARSP_2:th.28">Th36</a>, <a class="ref" href="parsp_1.html#T25" title="PARSP_1:th.25">PARSP_1:25</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:57_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="E4:57_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="firebrick">::  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="E5:57_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="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="parsp_2.html#T26" target="_self" title="PARSP_2:th.26">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="E7:57_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="E8:57_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="E9:57_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#T38" target="_self" title="PARSP_2:th.38">Th46</a></span>;<br/><a NAME="E10:57_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="#E7:57_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="E12:57_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="#E8:57_2"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="txt" href="#E2:57_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E1:57_2"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="ref" href="parsp_2.html#T11" target="_self" title="PARSP_2:th.11">Th17</a>, <a class="ref" href="parsp_2.html#T34" target="_self" title="PARSP_2:th.34">Th42</a></span>;<br/><a NAME="E13:57_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="#E4:57_2"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E12:57_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="parsp_2.html#T29" target="_self" title="PARSP_2:th.29">Th37</a></span>;<br/><a NAME="E14:57_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="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E7:57_2"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="txt" href="#E9:57_2"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E12:57_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="parsp_2.html#T41" target="_self" title="PARSP_2:th.41">Th49</a></span>;<br/><a NAME="E15:57_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="#E3:57"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E13:57_2"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="ref" href="parsp_2.html#T40" target="_self" title="PARSP_2:th.40">Th48</a></span>;<br/><a NAME="E16:57_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="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E3:57_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E7:57_2"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="txt" href="#E9:57_2"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E12:57_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="parsp_2.html#T29" target="_self" title="PARSP_2:th.29">Th37</a>, <a class="ref" href="parsp_2.html#T41" target="_self" title="PARSP_2:th.41">Th49</a></span>;<br/><a NAME="E17:57_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="#E2:57"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E12:57_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="parsp_2.html#T40" target="_self" title="PARSP_2:th.40">Th48</a></span>;<br/><span class="kw">hence </span><a NAME="E18:57_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="#E15:57_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="ref" href="parsp_2.html#T39" target="_self" title="PARSP_2:th.39">Th47</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:57_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="firebrick">::  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="E2:57_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="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="parsp_2.html#T40" target="_self" title="PARSP_2:th.40">Th48</a></span>;<br/><span class="kw">hence </span><a NAME="E3:57_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="#E3:57"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="parsp_2.html#T39" target="_self" title="PARSP_2:th.39">Th47</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="E7:57"/>
<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="#E4:57"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E5:57"><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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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