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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">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/><b>let </b><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:()"><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="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/>

















<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E48">A1</font></i>: 
(  <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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><i><font color="Green" title="E49">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_1"/>
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:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
(  <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> &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> )
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="parsp_2.html#T48" target="_self" title="PARSP_2:th.48">Th48</a></i>;<br/><b>hence </b><a NAME="E3:57_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>
 <b>by </b><i><a class="ref" href="parsp_2.html#T47" target="_self" title="PARSP_2:th.47">Th47</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>
<div><i><font color="Green" title="E50">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_2"/>
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:()"><i><font color="Red">::  thesis: </font></i></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_2"/><b>then </b>
( 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>  &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="c2">a</font>,<font color="Maroon" title="c3">p</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> &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> )
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="parsp_2.html#T41" target="_self" title="PARSP_2:th.41">Th41</a></i>;<br/><a NAME="E3:57_2"/><b>then </b>
(  <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> &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> )
 <b>by </b><i><a class="ref" href="parsp_2.html#T48" target="_self" title="PARSP_2:th.48">Th48</a></i>;<br/><b>hence </b><a NAME="E4:57_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>
 <b>by </b><i><a class="ref" href="parsp_2.html#T47" target="_self" title="PARSP_2:th.47">Th47</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_3"/><i><font color="Green" title="E51">A4</font></i>: 
( <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>  &amp; <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:()"><i><font color="Red">::  thesis: </font></i></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"/>
<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>
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="parsp_2.html#T34" target="_self" title="PARSP_2:th.34">Th34</a></i>;<br/><b>then consider </b><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><b> such that </b><br/><a NAME="E4:57_3"/><i><font color="Green" title="E52">A5</font></i>: 
( <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>  &amp; <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> &amp; <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> )
 <b>by </b><i><a class="ref" href="parsp_2.html#T46" target="_self" title="PARSP_2:th.46">Th46</a></i>;<br/><a NAME="E5:57_3"/><i><font color="Green" title="E53">A6</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="parsp_2.html#E4:57_3"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="parsp_2.html#D2" target="_self" title="PARSP_2:def.2">Def2</a></i>;<br/><a NAME="E6:57_3"/><i><font color="Green" title="E54">A7</font></i>: 
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> 
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="parsp_2.html#T36" target="_self" title="PARSP_2:th.36">Th36</a></i>;<br/><a NAME="E7:57_3"/><i><font color="Green" title="E55">A8</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="parsp_1.html#T42" title="PARSP_1:th.42">PARSP_1:42</a></i>;<br/><a NAME="E8:57_3"/>
<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>
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="parsp_2.html#T34" target="_self" title="PARSP_2:th.34">Th34</a></i>;<br/><a NAME="E9:57_3"/><b>then </b>
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c10">x</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="parsp_2.html#E4:57_3"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="parsp_2.html#E5:57_3"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="parsp_2.html#E6:57_3"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="parsp_2.html#E7:57_3"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="parsp_2.html#T17" target="_self" title="PARSP_2:th.17">Th17</a></i>;<br/><b>then consider </b><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><b> such that </b><br/><a NAME="E11:57_3"/><i><font color="Green" title="E56">A9</font></i>: 
 <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>
 <b>by </b><i><a class="ref" href="parsp_2.html#T42" target="_self" title="PARSP_2:th.42">Th42</a></i>;<br/><a NAME="E12:57_3"/><i><font color="Green" title="E57">A10</font></i>: 
 <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>
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="parsp_2.html#E4:57_3"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="parsp_2.html#E11:57_3"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="parsp_2.html#T49" target="_self" title="PARSP_2:th.49">Th49</a></i>;<br/><a NAME="E13:57_3"/><b>then </b>
( 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>  &amp; 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>  )
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57_3"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="parsp_2.html#E11:57_3"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="parsp_2.html#T37" target="_self" title="PARSP_2:th.37">Th37</a></i>;<br/><a NAME="E14:57_3"/><b>then </b>
(  <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> &amp;  <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> )
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:57"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="parsp_2.html#E11:57_3"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="parsp_2.html#E12:57_3"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="parsp_2.html#T48" target="_self" title="PARSP_2:th.48">Th48</a></i>;<br/><b>hence </b><a NAME="E15:57_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>
 <b>by </b><i><a class="ref" href="parsp_2.html#T47" target="_self" title="PARSP_2:th.47">Th47</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="E5:57"/>
<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>
 <b>by </b><i><a class="txt" href="parsp_2.html#E2:57"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="parsp_2.html#E3:57"><i><font color="Green" title="E50">A3</font></i></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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