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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</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">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st not <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  holds <br/>( not <font color="Olive" title="b1">u</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; not <font color="Olive" title="b2">v</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; not <font color="Olive" title="b3">w</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Olive" title="b3">w</font>,<font color="Olive" title="b1">u</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</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">u</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  implies ( not <font color="Maroon" title="c2">u</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; not <font color="Maroon" title="c3">v</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; not <font color="Maroon" title="c4">w</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:25"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
not <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 ; <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">u</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; not <font color="Maroon" title="c3">v</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; not <font color="Maroon" title="c4">w</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font> )</span><br/>

<a NAME="E2:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c4">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_1.html#T15" target="_self" title="ANPROJ_1:th.15">Th15</a></span>;<br/>
<a NAME="E3:25"/>
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="anproj_1.html#T15" target="_self" title="ANPROJ_1:th.15">Th15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:25"/>
( not <font color="Maroon" title="c2">u</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; not <font color="Maroon" title="c3">v</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; not <font color="Maroon" title="c4">w</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="struct_0.html#D12" title="STRUCT_0:def.12">STRUCT_0:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font> )</span><br/>

<span class="kw">thus </span><a NAME="E5:25"/>
( not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="anproj_1.html#T16" target="_self" title="ANPROJ_1:th.16">Th16</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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