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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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">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 not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> holds <br/> not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">d</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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>  &amp; <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="c5">d</font> implies  not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a>  )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
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> 
 <span class="kw">and </span><br/><a NAME="E2:36"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<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="c5">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a> </span><br/>

<div><span class="lab"><font color="Green" title="E31">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:36_1"/><span class="lab"><font color="Green" title="E33">A5</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="c5">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a> </span><br/><a NAME="E3:36_1"/>
( <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="c4">c</font>,<font color="Maroon" title="c2">a</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="parsp_2.html#T18" target="_self" title="PARSP_2:th.18">Th18</a>, <a class="ref" href="parsp_1.html#T42" title="PARSP_1:th.42">PARSP_1:42</a></span>;<br/><a NAME="E4:36_1"/><span class="kw">then </span>
not <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c2">a</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="#E1:36"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="parsp_2.html#T17" target="_self" title="PARSP_2:th.17">Th17</a></span>;<br/><a NAME="E5:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A6</font></span>: 
not <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</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="ref" href="parsp_2.html#T15" target="_self" title="PARSP_2:th.15">Th15</a></span>;<br/><a NAME="E6:36_1"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c2">a</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="parsp_1.html#T42" title="PARSP_1:th.42">PARSP_1:42</a></span>;<br/><a NAME="E7:36_1"/>
( <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="c5">d</font>,<font color="Maroon" title="c4">c</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">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="parsp_2.html#T18" target="_self" title="PARSP_2:th.18">Th18</a>, <a class="ref" href="parsp_1.html#T40" title="PARSP_1:th.40">PARSP_1:40</a></span>;<br/><span class="kw">hence </span><a NAME="E8:36_1"/>
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">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="#E2:36_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E5:36_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E6:36_1"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="parsp_2.html#T17" target="_self" title="PARSP_2:th.17">Th17</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="36_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:36_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:36_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="parsp_1.html#T40" title="PARSP_1:th.40">PARSP_1:40</a></span>;<br/><span class="kw">hence </span><a NAME="E3:36_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="parsp_2.html#D2" target="_self" title="PARSP_2:def.2">Def2</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="E5:36"/>
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">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="#E1:36"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E31">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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