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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v2</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v3</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">L</font> be    <a href="rlvect_2.html#M2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K8">}</a></span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v3</font> &amp; <font color="Maroon">v3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v1</font> )
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">L</font> is <a href="convex1.html#V2" target="_self">convex</a>
 ;<br/>

<a NAME="E3:46"/><i><font color="Green">E35</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K8">}</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<a NAME="E4:46"/><i><font color="Green">E37</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_1"/>
(  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v1</font></span><a href="domain_1.html#K6">}</a></span> or  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v2</font></span><a href="domain_1.html#K6">}</a></span> or  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v3</font></span><a href="domain_1.html#K6">}</a></span> or  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span> or  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K7">}</a></span> or  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K7">}</a></span> or  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K8">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="convex1.html#D2" target="_self">Def2</a>, <a class="ref" href="zfmisc_1.html#T142">ZFMISC_1:142</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v1</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:46_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E3:46_1_1_1"/>
( not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> &amp; not <font color="Maroon">v3</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:46_1_1_1"/><b>then </b>
( not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span>  &amp; not <font color="Maroon">v3</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span>  )
 <b>by </b><i><a class="ref" href="rlvect_2.html#D6">RLVECT_2:def 6</a></i>;<br/><a NAME="E5:46_1_1_1"/><b>then </b>
( <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><b>hence </b><a NAME="E6:46_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="convex1.html#E1:46"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_2"/><i><font color="Green">E47</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v2</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:46_1_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E3:46_1_1_2"/>
( not <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> &amp; not <font color="Maroon">v3</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:46_1_1_2"/><b>then </b>
( not <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span>  &amp; not <font color="Maroon">v3</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span>  )
 <b>by </b><i><a class="ref" href="rlvect_2.html#D6">RLVECT_2:def 6</a></i>;<br/><a NAME="E5:46_1_1_2"/><b>then </b>
( <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><b>hence </b><a NAME="E6:46_1_1_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="convex1.html#E1:46"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_3"><b>suppose </b></a><a NAME="E1:46_1_1_3"/><i><font color="Green">E52</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v3</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:46_1_1_3"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E3:46_1_1_3"/>
( not <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> &amp; not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:46_1_1_3"/><b>then </b>
( not <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span>  &amp; not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span>  )
 <b>by </b><i><a class="ref" href="rlvect_2.html#D6">RLVECT_2:def 6</a></i>;<br/><a NAME="E5:46_1_1_3"/><b>then </b>
( <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><b>hence </b><a NAME="E6:46_1_1_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="convex1.html#E1:46"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_4"><b>suppose </b></a><a NAME="E1:46_1_1_4"/><i><font color="Green">E55</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><div class="add"><a NAME="E2:46_1_1_4"/><b>then </b>
not <font color="Maroon">v3</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E3:46_1_1_4"/><b>then </b>
not <font color="Maroon">v3</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span> 
 <b>by </b><i><a class="ref" href="rlvect_2.html#D6">RLVECT_2:def 6</a></i>;<br/><a NAME="E4:46_1_1_4"/><b>then </b>
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E5:46_1_1_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_5"><b>suppose </b></a><a NAME="E1:46_1_1_5"/><i><font color="Green">E60</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><div class="add"><a NAME="E2:46_1_1_5"/><b>then </b>
not <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E3:46_1_1_5"/><b>then </b>
not <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span> 
 <b>by </b><i><a class="ref" href="rlvect_2.html#D6">RLVECT_2:def 6</a></i>;<br/><a NAME="E4:46_1_1_5"/><b>then </b>
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E5:46_1_1_5"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_6"><b>suppose </b></a><a NAME="E1:46_1_1_6"/><i><font color="Green">E61</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><div class="add"><a NAME="E2:46_1_1_6"/><b>then </b>
not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E3:46_1_1_6"/><b>then </b>
not <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span> 
 <b>by </b><i><a class="ref" href="rlvect_2.html#D6">RLVECT_2:def 6</a></i>;<br/><a NAME="E4:46_1_1_6"/><b>then </b>
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E5:46_1_1_6"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_7"><b>suppose </b></a><a NAME="E1:46_1_1_7"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">v3</font></span><a href="domain_1.html#K8">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_1_1_7"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:46"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v3</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


</div>
