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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">b</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">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> holds <br/> ex <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FdSp</font> st <br/>( <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="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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"> ( <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> implies  ex <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FdSp</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">c</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> ) )</span><br/>





<b>assume </b><a NAME="E1:53"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FdSp</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">c</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c4">p</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="E3:53"/><i><font color="Green" title="E44">A1</font></i>: 
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">p</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#T20" target="_self" title="PARSP_2:th.20">Th20</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">q</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="E5:53"/><i><font color="Green" title="E45">A2</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">p</font>,<font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="parsp_2.html#E3:53"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="parsp_2.html#T42" target="_self" title="PARSP_2:th.42">Th42</a></i>;<br/>
<a NAME="E6:53"/>
not <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">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#E5:53"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="parsp_2.html#T36" target="_self" title="PARSP_2:th.36">Th36</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">c</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="E8:53"/><i><font color="Green" title="E46">A3</font></i>: 
 <a href="parsp_2.html#R2" title="PARSP_2:pred.2">parallelogram</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">c</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="E9:53"/>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</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="c6">c</font> &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">p</font>,<font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">c</font> )
 
<b>by </b><i><a class="txt" href="parsp_2.html#E5:53"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="parsp_2.html#E8:53"><i><font color="Green" title="E46">A3</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_2.html#T34" target="_self" title="PARSP_2:th.34">Th34</a></i>;<br/>
<a NAME="E10:53"/><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="c3">b</font>,<font color="Maroon" title="c6">c</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="c6">c</font> &amp; not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">q</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">p</font> &amp; <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">c</font> )
 
<b>by </b><i><a class="txt" href="parsp_2.html#E5:53"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="parsp_2.html#T44" target="_self" title="PARSP_2:th.44">Th44</a>, <a class="ref" href="parsp_1.html#T43" title="PARSP_1:th.43">PARSP_1:43</a></i>;<br/>
<a NAME="E11:53"/><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="c6">c</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="c6">c</font> &amp; not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">q</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">p</font> &amp; <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">q</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">p</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="E12:53"/><b>then </b>
( <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</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="c6">c</font> )
 
<b>by </b><i><a class="ref" href="parsp_2.html#D2" target="_self" title="PARSP_2:def.2">Def2</a></i>;<br/>
<a NAME="E13:53"/><b>then </b>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c6">c</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="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="ref" href="parsp_2.html#T15" target="_self" title="PARSP_2:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E14:53"/>
 ex <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FdSp</font> st <br/>( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">c</font> <a href="parsp_2.html#R1" title="PARSP_2:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 ; <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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