<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">V</font> being  <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a><br/> for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">V</font>  st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b2">A</font> &amp; <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  holds <br/><font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b1">A</font> &amp; <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  holds <br/><font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">A</font> &amp; <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  implies <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:90"/><span class="lab"><font color="Green" title="E66">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E2:90"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
<span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">A1</font> = <font color="Maroon" title="c3">A</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E66">A1</font></span></a></span>;<br/>
<a NAME="E4:90"/><span class="lab"><font color="Green" title="E68">A3</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">L</font></span>)</span> where <font color="Olive" title="b1">L</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c4">A1</font> : <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c2">V</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex3.html#T5" title="CONVEX3:th.5">CONVEX3:5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:90"/><span class="lab"><font color="Green" title="E69">A4</font></span>: 
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</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/>

<span class="kw">consider </span><font color="Maroon" title="c5">L</font> being   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c4">A1</font><span class="kw"> such that </span><br/><a NAME="E7:90"/><span class="lab"><font color="Green" title="E70">A5</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 <span class="kw">and </span><br/><a NAME="E8:90"/>
<font color="Maroon" title="c5">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E4:90"><span class="lab"><font color="Green" title="E68">A3</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">C</font> =  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font>;<br/>
<a NAME="E9:90"/><span class="lab"><font color="Green" title="E71">A6</font></span>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D8" title="RLVECT_2:def.8">RLVECT_2:def 8</a></span>;<br/>
<a NAME="E10:90"/>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T21" title="CONVEX1:th.21">CONVEX1:21</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E12:90"/><span class="lab"><font color="Green" title="E72">A7</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c7">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:90"><span class="lab"><font color="Green" title="E72">A7</font></span></a></span>;<br/>
<a NAME="E14:90"/><span class="lab"><font color="Green" title="E73">A8</font></span>: 
 <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T62" target="_self" title="RLAFFIN1:th.62">Th62</a></span>;<br/>
<a NAME="E15:90"/>
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:90_1"/><span class="lab"><font color="Green" title="E74">A9</font></span>: 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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="E2:90_1"/><span class="kw">then </span>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:90"><span class="lab"><font color="Green" title="E72">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T66" title="ZFMISC_1:th.66">ZFMISC_1:66</a></span>;<br/>
<a NAME="E3:90_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A10</font></span>: 
<font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:90"><span class="lab"><font color="Green" title="E73">A8</font></span></a>, <a class="ref" href="rlaffin1.html#T32" target="_self" title="RLAFFIN1:th.32">Th32</a></span>;<br/>
<a NAME="E4:90_1"/>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:90"><span class="lab"><font color="Green" title="E72">A7</font></span></a>, <a class="txt" href="#E1:90_1"><span class="lab"><font color="Green" title="E74">A9</font></span></a>, <a class="ref" href="rlvect_2.html#T53" title="RLVECT_2:th.53">RLVECT_2:53</a>, <a class="ref" href="zfmisc_1.html#T66" title="ZFMISC_1:th.66">ZFMISC_1:66</a></span>;<br/>
<a NAME="E5:90_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90"><span class="lab"><font color="Green" title="E70">A5</font></span></a>, <a class="txt" href="#E3:90_1"><span class="lab"><font color="Green" title="E75">A10</font></span></a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:90_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E69">A4</font></span></a>, <a class="txt" href="#E9:90"><span class="lab"><font color="Green" title="E71">A6</font></span></a>, <a class="txt" href="#E12:90"><span class="lab"><font color="Green" title="E72">A7</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/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:90"/><span class="lab"><font color="Green" title="E74">A11</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c9">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:90"><span class="lab"><font color="Green" title="E74">A11</font></span></a></span>;<br/>
<a NAME="E19:90"/><span class="lab"><font color="Green" title="E75">A12</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:90"><span class="lab"><font color="Green" title="E74">A11</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E20:90"/><span class="kw">then </span>
<font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T28" title="RLVECT_2:th.28">RLVECT_2:28</a></span>;<br/>
<a NAME="E21:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A13</font></span>: 
<font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T62" target="_self" title="RLAFFIN1:th.62">Th62</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">mm</font> =  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">q</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">Lq</font> being    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span><span class="kw"> such that </span><br/><a NAME="E24:90"/><span class="lab"><font color="Green" title="E77">A14</font></span>: 
<font color="Maroon" title="c12">Lq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">mm</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="rlvect_4.html#S2" title="RLVECT_4:sch.2">RLVECT_4:sch 2</a>();<br/></span>
<a NAME="E25:90"/><span class="lab"><font color="Green" title="E78">A15</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:90"><span class="lab"><font color="Green" title="E74">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/>
<a NAME="E26:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A16</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c10">q</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="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T35" title="RLVECT_1:th.35">RLVECT_1:35</a></span>;<br/>
<a NAME="E27:90"/><span class="lab"><font color="Green" title="E80">A17</font></span>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c12">Lq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D8" title="RLVECT_2:def.8">RLVECT_2:def 8</a></span>;<br/>
<a NAME="E28:90"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:90"><span class="lab"><font color="Green" title="E71">A6</font></span></a>, <a class="txt" href="#E19:90"><span class="lab"><font color="Green" title="E75">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<a NAME="E29:90"/><span class="kw">then </span>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c12">Lq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:90"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E30:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A18</font></span>: 
<font color="Maroon" title="c12">Lq</font> is    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D8" title="RLVECT_2:def.8">RLVECT_2:def 8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">Lp</font> being    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span><span class="kw"> such that </span><br/><a NAME="E32:90"/><span class="lab"><font color="Green" title="E82">A19</font></span>: 
<font color="Maroon" title="c13">Lp</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">mm</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="rlvect_4.html#S2" title="RLVECT_4:sch.2">RLVECT_4:sch 2</a>();<br/></span>
<a NAME="E33:90"/><span class="lab"><font color="Green" title="E83">A20</font></span>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c13">Lp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D8" title="RLVECT_2:def.8">RLVECT_2:def 8</a></span>;<br/>
<a NAME="E34:90"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:90"><span class="lab"><font color="Green" title="E71">A6</font></span></a>, <a class="txt" href="#E12:90"><span class="lab"><font color="Green" title="E72">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<a NAME="E35:90"/><span class="kw">then </span>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c13">Lp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:90"><span class="lab"><font color="Green" title="E83">A20</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E36:90"/><span class="kw">then </span>
<font color="Maroon" title="c13">Lp</font> is    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D8" title="RLVECT_2:def.8">RLVECT_2:def 8</a></span>;<br/>
<a NAME="E37:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A21</font></span>: 
<font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font> is    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:90"><span class="lab"><font color="Green" title="E81">A18</font></span></a>, <a class="ref" href="rlvect_2.html#T81" title="RLVECT_2:th.81">RLVECT_2:81</a></span>;<br/>
<a NAME="E38:90"/><span class="kw">then </span>
 <a href="rlvect_2.html#K9" title="RLVECT_2:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span> is    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T76" title="RLVECT_2:th.76">RLVECT_2:76</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">Lpq</font> = <font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span>, <font color="Maroon" title="c15">L1pq</font> = <font color="Maroon" title="c5">L</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span> as    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:90"><span class="lab"><font color="Green" title="E84">A21</font></span></a>, <a class="ref" href="rlvect_2.html#T59" title="RLVECT_2:th.59">RLVECT_2:59</a></span>;<br/>
<span class="lab"><font color="Green" title="E85">A22</font></span>:  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c15">L1pq</font> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_3.html#T4" title="RLVECT_3:th.4">RLVECT_3:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D14" title="RLVECT_1:def.14">RLVECT_1:def 14</a></span>
;<br/>
<a NAME="E41:90"/>
<font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:90"><span class="lab"><font color="Green" title="E72">A7</font></span></a>, <a class="ref" href="rlvect_2.html#T28" title="RLVECT_2:th.28">RLVECT_2:28</a></span>;<br/>
<a NAME="E42:90"/><span class="kw">then </span>
<font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T62" target="_self" title="RLAFFIN1:th.62">Th62</a></span>;<br/>
<a NAME="E43:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A23</font></span>: 
<font color="Maroon" title="c11">mm</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:90"><span class="lab"><font color="Green" title="E76">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></span>;<br/>
<a NAME="E44:90"/><span class="lab"><font color="Green" title="E87">A24</font></span>: 
for <font color="Olive" title="b1">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font> holds  <font color="Maroon" title="c15">L1pq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">L1pq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="lab"><font color="Green" title="E88">A25</font></span>: <font color="Maroon" title="c15">L1pq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T79" title="RLVECT_2:th.79">RLVECT_2:79</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Lq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T79" title="RLVECT_2:th.79">RLVECT_2:79</a></span>
;<br/>
<a NAME="E2:90_3"/><span class="lab"><font color="Green" title="E89">A26</font></span>: 
<font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T62" target="_self" title="RLAFFIN1:th.62">Th62</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_3_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:90_3_2"/>
( <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> or <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> or ( <font color="Maroon" title="c16">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c16">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_3_2_1"><span class="kw">suppose </span></a><a NAME="E1:90_3_2_1"/><span class="lab"><font color="Green" title="E90">A27</font></span>: 
<font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">L1pq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:90_3_2_1"/><span class="kw">then </span>
not <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c13">Lp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:90"><span class="lab"><font color="Green" title="E78">A15</font></span></a>, <a class="txt" href="#E33:90"><span class="lab"><font color="Green" title="E83">A20</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:90_3_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">Lp</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><span class="kw">hence </span><a NAME="E4:90_3_2_1"/>
<font color="Maroon" title="c15">L1pq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:90"><span class="lab"><font color="Green" title="E86">A23</font></span></a>, <a class="txt" href="#E24:90"><span class="lab"><font color="Green" title="E77">A14</font></span></a>, <a class="txt" href="#E1:90_3"><span class="lab"><font color="Green" title="E88">A25</font></span></a>, <a class="txt" href="#E2:90_3"><span class="lab"><font color="Green" title="E89">A26</font></span></a>, <a class="txt" href="#E1:90_3_2_1"><span class="lab"><font color="Green" title="E90">A27</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_3_2_2"><span class="kw">suppose </span></a><a NAME="E1:90_3_2_2"/><span class="lab"><font color="Green" title="E90">A28</font></span>: 
<font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">L1pq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:90_3_2_2"/>
<font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">mm</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/><a NAME="E3:90_3_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A29</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">mm</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">mm</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">mm</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E4:90_3_2_2"/>
not <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c12">Lq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:90"><span class="lab"><font color="Green" title="E78">A15</font></span></a>, <a class="txt" href="#E27:90"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="txt" href="#E1:90_3_2_2"><span class="lab"><font color="Green" title="E90">A28</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E5:90_3_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">Lq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><span class="kw">hence </span><a NAME="E6:90_3_2_2"/>
<font color="Maroon" title="c15">L1pq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:90"><span class="lab"><font color="Green" title="E82">A19</font></span></a>, <a class="txt" href="#E1:90_3"><span class="lab"><font color="Green" title="E88">A25</font></span></a>, <a class="txt" href="#E1:90_3_2_2"><span class="lab"><font color="Green" title="E90">A28</font></span></a>, <a class="txt" href="#E3:90_3_2_2"><span class="lab"><font color="Green" title="E91">A29</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_3_2_3"><span class="kw">suppose </span></a><a NAME="E1:90_3_2_3"/><span class="lab"><font color="Green" title="E90">A30</font></span>: 
( <font color="Maroon" title="c16">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c16">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">L1pq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:90_3_2_3"/><span class="kw">then </span>
not <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c12">Lq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:90"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:90_3_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A31</font></span>: 
<font color="Maroon" title="c12">Lq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E4:90_3_2_3"/>
not <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c13">Lp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:90"><span class="lab"><font color="Green" title="E83">A20</font></span></a>, <a class="txt" href="#E1:90_3_2_3"><span class="lab"><font color="Green" title="E90">A30</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E5:90_3_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">Lp</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><span class="kw">hence </span><a NAME="E6:90_3_2_3"/>
<font color="Maroon" title="c15">L1pq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_3"><span class="lab"><font color="Green" title="E88">A25</font></span></a>, <a class="txt" href="#E3:90_3_2_3"><span class="lab"><font color="Green" title="E91">A31</font></span></a>, <a class="ref" href="rlaffin1.html#T62" target="_self" title="RLAFFIN1:th.62">Th62</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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c13">Lp</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c12">Lq</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_3.html#T4" title="RLVECT_3:th.4">RLVECT_3:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">mm</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c12">Lq</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:90"><span class="lab"><font color="Green" title="E82">A19</font></span></a>, <a class="ref" href="rlvect_2.html#T50" title="RLVECT_2:th.50">RLVECT_2:50</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">mm</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">mm</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:90"><span class="lab"><font color="Green" title="E77">A14</font></span></a>, <a class="ref" href="rlvect_2.html#T50" title="RLVECT_2:th.50">RLVECT_2:50</a></span>
<br/>.= 
<font color="Maroon" title="c11">mm</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c10">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T48" title="RLVECT_1:th.48">RLVECT_1:48</a></span>
;<br/>
<a NAME="E46:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A32</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span> <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="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:90"><span class="lab"><font color="Green" title="E86">A23</font></span></a>, <a class="txt" href="#E26:90"><span class="lab"><font color="Green" title="E79">A16</font></span></a>, <a class="ref" href="rlvect_1.html#T24" title="RLVECT_1:th.24">RLVECT_1:24</a></span>;<br/>
<span class="lab"><font color="Green" title="E89">A33</font></span>:  <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c13">Lp</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c12">Lq</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T36" target="_self" title="RLAFFIN1:th.36">Th36</a></span>
<br/>.= 
<font color="Maroon" title="c11">mm</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c12">Lq</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:90"><span class="lab"><font color="Green" title="E82">A19</font></span></a>, <a class="txt" href="#E33:90"><span class="lab"><font color="Green" title="E83">A20</font></span></a>, <a class="ref" href="rlaffin1.html#T32" target="_self" title="RLAFFIN1:th.32">Th32</a></span>
<br/>.= 
<font color="Maroon" title="c11">mm</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">mm</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:90"><span class="lab"><font color="Green" title="E77">A14</font></span></a>, <a class="txt" href="#E27:90"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="ref" href="rlaffin1.html#T32" target="_self" title="RLAFFIN1:th.32">Th32</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/>
<a NAME="E48:90"/><span class="lab"><font color="Green" title="E90">A34</font></span>: 
for <font color="Olive" title="b1">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font> holds  <font color="Maroon" title="c14">Lpq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">Lpq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="lab"><font color="Green" title="E91">A35</font></span>: <font color="Maroon" title="c14">Lpq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D12" title="RLVECT_2:def.12">RLVECT_2:def 12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">Lq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T79" title="RLVECT_2:th.79">RLVECT_2:79</a></span>
;<br/>
<a NAME="E2:90_6"/><span class="lab"><font color="Green" title="E92">A36</font></span>: 
<font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T62" target="_self" title="RLAFFIN1:th.62">Th62</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_6_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:90_6_2"/>
( <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> or <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> or ( <font color="Maroon" title="c16">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c16">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_6_2_1"><span class="kw">suppose </span></a><a NAME="E1:90_6_2_1"/><span class="lab"><font color="Green" title="E93">A37</font></span>: 
<font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">Lpq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:90_6_2_1"/><span class="kw">then </span>
not <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c12">Lq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:90"><span class="lab"><font color="Green" title="E78">A15</font></span></a>, <a class="txt" href="#E27:90"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:90_6_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">Lpq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">mm</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:90"><span class="lab"><font color="Green" title="E82">A19</font></span></a>, <a class="txt" href="#E1:90_6"><span class="lab"><font color="Green" title="E91">A35</font></span></a>, <a class="txt" href="#E1:90_6_2_1"><span class="lab"><font color="Green" title="E93">A37</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:90_6_2_1"/>
<font color="Maroon" title="c14">Lpq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:90"><span class="lab"><font color="Green" title="E86">A23</font></span></a>, <a class="txt" href="#E2:90_6"><span class="lab"><font color="Green" title="E92">A36</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_6_2_2"><span class="kw">suppose </span></a><a NAME="E1:90_6_2_2"/><span class="lab"><font color="Green" title="E93">A38</font></span>: 
<font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">Lpq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:90_6_2_2"/>
<font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">q</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">mm</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/><a NAME="E3:90_6_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A39</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">q</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">mm</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">mm</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">mm</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E4:90_6_2_2"/>
not <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c13">Lp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:90"><span class="lab"><font color="Green" title="E78">A15</font></span></a>, <a class="txt" href="#E33:90"><span class="lab"><font color="Green" title="E83">A20</font></span></a>, <a class="txt" href="#E1:90_6_2_2"><span class="lab"><font color="Green" title="E93">A38</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E5:90_6_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">Lp</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><span class="kw">hence </span><a NAME="E6:90_6_2_2"/>
<font color="Maroon" title="c14">Lpq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:90"><span class="lab"><font color="Green" title="E77">A14</font></span></a>, <a class="txt" href="#E1:90_6"><span class="lab"><font color="Green" title="E91">A35</font></span></a>, <a class="txt" href="#E1:90_6_2_2"><span class="lab"><font color="Green" title="E93">A38</font></span></a>, <a class="txt" href="#E3:90_6_2_2"><span class="lab"><font color="Green" title="E94">A39</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_6_2_3"><span class="kw">suppose </span></a><a NAME="E1:90_6_2_3"/><span class="lab"><font color="Green" title="E93">A40</font></span>: 
( <font color="Maroon" title="c16">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c16">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">Lpq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:90_6_2_3"/><span class="kw">then </span>
not <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c12">Lq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:90"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:90_6_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A41</font></span>: 
<font color="Maroon" title="c12">Lq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E4:90_6_2_3"/>
not <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c13">Lp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:90"><span class="lab"><font color="Green" title="E83">A20</font></span></a>, <a class="txt" href="#E1:90_6_2_3"><span class="lab"><font color="Green" title="E93">A40</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E5:90_6_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">Lp</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><span class="kw">hence </span><a NAME="E6:90_6_2_3"/>
<font color="Maroon" title="c14">Lpq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">v</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_6"><span class="lab"><font color="Green" title="E91">A35</font></span></a>, <a class="txt" href="#E3:90_6_2_3"><span class="lab"><font color="Green" title="E94">A41</font></span></a>, <a class="ref" href="rlaffin1.html#T62" target="_self" title="RLAFFIN1:th.62">Th62</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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
 <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c15">L1pq</font> = 
<span class="p1">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T36" target="_self" title="RLAFFIN1:th.36">Th36</a></span>
<br/>.= 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:90"><span class="lab"><font color="Green" title="E89">A33</font></span></a>, <a class="ref" href="rlaffin1.html#T62" target="_self" title="RLAFFIN1:th.62">Th62</a></span>
;<br/>
<a NAME="E50:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A42</font></span>: 
<font color="Maroon" title="c15">L1pq</font> is  <a href="convex1.html#V2" title="CONVEX1:attr.2">convex</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:90"><span class="lab"><font color="Green" title="E87">A24</font></span></a>, <a class="ref" href="rlaffin1.html#T62" target="_self" title="RLAFFIN1:th.62">Th62</a></span>;<br/>
<a NAME="E51:90"/><span class="kw">then </span>
<font color="Maroon" title="c15">L1pq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex3.html#D1" title="CONVEX3:def.1">CONVEX3:def 1</a></span>;<br/>
<a NAME="E52:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A43</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c15">L1pq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c4">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90"><span class="lab"><font color="Green" title="E68">A3</font></span></a>, <a class="txt" href="#E50:90"><span class="lab"><font color="Green" title="E91">A42</font></span></a></span>;<br/>
 <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c14">Lpq</font> = 
<span class="p1">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T34" target="_self" title="RLAFFIN1:th.34">Th34</a></span>
<br/>.= 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:90"><span class="lab"><font color="Green" title="E89">A33</font></span></a>, <a class="ref" href="rlaffin1.html#T62" target="_self" title="RLAFFIN1:th.62">Th62</a></span>
;<br/>
<a NAME="E54:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A44</font></span>: 
<font color="Maroon" title="c14">Lpq</font> is  <a href="convex1.html#V2" title="CONVEX1:attr.2">convex</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:90"><span class="lab"><font color="Green" title="E90">A34</font></span></a>, <a class="ref" href="rlaffin1.html#T62" target="_self" title="RLAFFIN1:th.62">Th62</a></span>;<br/>
<a NAME="E55:90"/><span class="kw">then </span>
<font color="Maroon" title="c14">Lpq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex3.html#D1" title="CONVEX3:def.1">CONVEX3:def 1</a></span>;<br/>
<a NAME="E56:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A45</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c14">Lpq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90"><span class="lab"><font color="Green" title="E68">A3</font></span></a>, <a class="txt" href="#E54:90"><span class="lab"><font color="Green" title="E93">A44</font></span></a></span>;<br/>
<a NAME="E57:90"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T25" title="RLVECT_1:th.25">RLVECT_1:25</a></span>;<br/>
<a NAME="E58:90"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span> <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="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:90"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="ref" href="rlvect_1.html#T31" title="RLVECT_1:th.31">RLVECT_1:31</a></span>;<br/>
<a NAME="E59:90"/><span class="kw">then </span>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c15">L1pq</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90"><span class="lab"><font color="Green" title="E70">A5</font></span></a>, <a class="txt" href="#E40:90"><span class="lab"><font color="Green" title="E85">A22</font></span></a>, <a class="ref" href="rlvect_1.html#T22" title="RLVECT_1:th.22">RLVECT_1:22</a></span>;<br/>
<a NAME="E60:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A46</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c15">L1pq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:90"><span class="lab"><font color="Green" title="E92">A43</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/>
<a NAME="E61:90"/>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c14">Lpq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_3.html#T1" title="RLVECT_3:th.1">RLVECT_3:1</a></span>;<br/>
<a NAME="E62:90"/><span class="kw">then </span>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c14">Lpq</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90"><span class="lab"><font color="Green" title="E70">A5</font></span></a>, <a class="txt" href="#E46:90"><span class="lab"><font color="Green" title="E88">A32</font></span></a>, <a class="ref" href="rlvect_1.html#T22" title="RLVECT_1:th.22">RLVECT_1:22</a></span>;<br/>
<a NAME="E63:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A47</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c14">Lpq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:90"><span class="lab"><font color="Green" title="E94">A45</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c14">Lpq</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c15">L1pq</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:90"><span class="lab"><font color="Green" title="E85">A22</font></span></a>, <a class="ref" href="rlvect_3.html#T1" title="RLVECT_3:th.1">RLVECT_3:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p5">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p0">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p5">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p0">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p5">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p0">(<span class="default"><font color="Maroon" title="c13">Lp</font> <a href="rlvect_2.html#K10" title="RLVECT_2:func.10">-</a> <font color="Maroon" title="c12">Lq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></span>
<br/>.= 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
;<br/>
<a NAME="E65:90"/><span class="kw">then </span>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:90"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E60:90"><span class="lab"><font color="Green" title="E95">A46</font></span></a>, <a class="txt" href="#E63:90"><span class="lab"><font color="Green" title="E96">A47</font></span></a>, <a class="ref" href="convex1.html#D2" title="CONVEX1:def.2">CONVEX1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E66:90"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90"><span class="lab"><font color="Green" title="E70">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</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/>

<span class="kw">set </span><font color="Maroon" title="c16">mp</font> =  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">p</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">mq</font> =  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">q</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">q</font></span>)</span></span>)</span>;<br/>

</div>
