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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</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">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  holds <br/>for <font color="Olive" title="b4">u</font>, <font color="Olive" title="b5">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b4">u</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Olive" title="b5">v</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Olive" title="b4">u</font>,<font color="Olive" title="b5">v</font> holds <br/> ex <font color="Olive" title="b6">y</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="b4">u</font>,<font color="Olive" title="b5">v</font>,<font color="Olive" title="b6">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> </span><br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  implies for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">u</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Olive" title="b2">v</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</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> holds <br/> ex <font color="Olive" title="b3">y</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">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )</span><br/>







<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E23">A1</font></i>: 
not <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</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">u</font>, <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">u</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Olive" title="b2">v</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</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> holds <br/> ex <font color="Olive" title="b3">y</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">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> </span><br/>

<b>let </b><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">v</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">u</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c6">v</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font> implies  ex <font color="Olive" title="b1">y</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="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )</span><br/>



<b>assume </b><a NAME="E2:27"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c5">u</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c6">v</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> </span><br/>

<b>assume </b><a NAME="E3:27"/>
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:27"/><b>then </b>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c2">p</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c3">q</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">r</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 
;<br/>
<b>hence </b><a NAME="E5:27"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:27"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="anproj_1.html#E2:27"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="anproj_1.html#T19" target="_self" title="ANPROJ_1:th.19">Th19</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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