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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">d</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">r</font>,<font color="Olive" title="b2">s</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">d</font> holds <br/><font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">d</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</font>, <font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">d</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"> ( <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> implies <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:83"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>
 <span class="kw">and </span><br/><a NAME="E2:83"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</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="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/>

<div><span class="lab"><font color="Green" title="E60">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:83_1"/>
<font color="Maroon" title="c2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">and </span><br/><a NAME="E2:83_1"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
( not <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">a</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</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="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/><a NAME="E3:83_1"/>
(  <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp;  <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E2:83_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="parsp_2.html#T60" target="_self" title="PARSP_2:th.60">Th60</a></span>;<br/><span class="kw">hence </span><a NAME="E4:83_1"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="parsp_2.html#D4" target="_self" title="PARSP_2:def.4">Def4</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="E61">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:83_2"/><span class="lab"><font color="Green" title="E62">A6</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">and </span><br/><a NAME="E2:83_2"/><span class="lab"><font color="Green" title="E63">A7</font></span>: 
<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</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="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/><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="E4:83_2"/><span class="lab"><font color="Green" title="E64">A8</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">r</font>,<font color="Maroon" title="c3">s</font>
 <span class="kw">and </span><br/><a NAME="E5:83_2"/><span class="lab"><font color="Green" title="E65">A9</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">a</font>,<font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E1:83_2"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="ref" href="parsp_2.html#D4" target="_self" title="PARSP_2:def.4">Def4</a></span>;<br/><a NAME="E6:83_2"/>
 <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="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E2:83_2"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="txt" href="#E4:83_2"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="ref" href="parsp_2.html#T62" target="_self" title="PARSP_2:th.62">Th62</a></span>;<br/><span class="kw">hence </span><a NAME="E7:83_2"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_2"><span class="lab"><font color="Green" title="E65">A9</font></span></a>, <a class="ref" href="parsp_2.html#D4" target="_self" title="PARSP_2:def.4">Def4</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="E62">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:83_3"/><span class="lab"><font color="Green" title="E63">A11</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">and </span><br/><a NAME="E2:83_3"/><span class="lab"><font color="Green" title="E64">A12</font></span>: 
<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">a</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="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/><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="E4:83_3"/><span class="lab"><font color="Green" title="E65">A13</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">r</font>,<font color="Maroon" title="c3">s</font>
 <span class="kw">and </span><br/><a NAME="E5:83_3"/><span class="lab"><font color="Green" title="E66">A14</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="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E1:83_3"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="ref" href="parsp_2.html#D4" target="_self" title="PARSP_2:def.4">Def4</a></span>;<br/><a NAME="E6:83_3"/>
 <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">a</font>,<font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:83_3"><span class="lab"><font color="Green" title="E64">A12</font></span></a>, <a class="txt" href="#E4:83_3"><span class="lab"><font color="Green" title="E65">A13</font></span></a>, <a class="ref" href="parsp_2.html#T62" target="_self" title="PARSP_2:th.62">Th62</a></span>;<br/><span class="kw">hence </span><a NAME="E7:83_3"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_3"><span class="lab"><font color="Green" title="E66">A14</font></span></a>, <a class="ref" href="parsp_2.html#D4" target="_self" title="PARSP_2:def.4">Def4</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="83_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:83_4"/>
<font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/><a NAME="E2:83_4"/><span class="kw">then </span>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="parsp_2.html#T55" target="_self" title="PARSP_2:th.55">Th55</a></span>;<br/><span class="kw">hence </span><a NAME="E3:83_4"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="parsp_2.html#D4" target="_self" title="PARSP_2:def.4">Def4</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="E7:83"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="parsp_2.html#R3" title="PARSP_2:pred.3">congr</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E60">A3</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/>


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