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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>;<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> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FdSp</font>;<br/>









<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E32">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">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 ;<br/>

<a NAME="E2:41"/><b>then </b><i><font color="Green" title="E33">A2</font></i>: 
not <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> 
 
<b>by </b><i><a class="ref" href="parsp_2.html#D3" target="_self" title="PARSP_2:def.3">Def3</a></i>;<br/>
<div><i><font color="Green" title="E34">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 ;<br/><a NAME="E2:41_1"/><b>then </b>
<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="c2">a</font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:41"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="parsp_2.html#D3" target="_self" title="PARSP_2:def.3">Def3</a></i>;<br/><a NAME="E3:41_1"/><b>then </b>
( <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="c2">a</font>,<font color="Maroon" title="c4">c</font> &amp; not <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>  )
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:41"><i><font color="Green" title="E32">A1</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_1.html#T40" title="PARSP_1:th.40">PARSP_1:40</a></i>;<br/><b>hence </b><a NAME="E4:41_1"/>
contradiction
 <b>by </b><i><a class="ref" href="parsp_2.html#D2" target="_self" title="PARSP_2:def.2">Def2</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E35">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_2"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 ;<br/><a NAME="E2:41_2"/><b>then </b>
<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="c3">b</font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:41"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="parsp_2.html#D3" target="_self" title="PARSP_2:def.3">Def3</a></i>;<br/><a NAME="E3:41_2"/><b>then </b>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="ref" href="parsp_1.html#T40" title="PARSP_1:th.40">PARSP_1:40</a></i>;<br/><a NAME="E4:41_2"/><b>then </b>
( <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="c2">a</font>,<font color="Maroon" title="c4">c</font> &amp; not <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>  )
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:41"><i><font color="Green" title="E32">A1</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_1.html#T41" title="PARSP_1:th.41">PARSP_1:41</a></i>;<br/><b>hence </b><a NAME="E5:41_2"/>
contradiction
 <b>by </b><i><a class="ref" href="parsp_2.html#D2" target="_self" title="PARSP_2:def.2">Def2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_3"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 ;<br/><a NAME="E2:41_3"/><b>then </b>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:41"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="parsp_2.html#D3" target="_self" title="PARSP_2:def.3">Def3</a></i>;<br/><a NAME="E3:41_3"/><b>then </b>
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="parsp_1.html#T40" title="PARSP_1:th.40">PARSP_1:40</a></i>;<br/><a NAME="E4:41_3"/><b>then </b>
( <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="c2">a</font>,<font color="Maroon" title="c4">c</font> &amp; not <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>  )
 <b>by </b><i><a class="txt" href="parsp_2.html#E1:41"><i><font color="Green" title="E32">A1</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_1.html#T41" title="PARSP_1:th.41">PARSP_1:41</a></i>;<br/><b>hence </b><a NAME="E5:41_3"/>
contradiction
 <b>by </b><i><a class="ref" href="parsp_2.html#D2" target="_self" title="PARSP_2:def.2">Def2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:41"/>
( <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="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c4">c</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="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="txt" href="parsp_2.html#E2:41"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="parsp_2.html#E3:41"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="parsp_2.html#E4:41"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="parsp_2.html#T18" target="_self" title="PARSP_2:th.18">Th18</a></i>;<br/>


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