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<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">I</font> being  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <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> <font color="Olive" title="b2">I</font> holds <br/><span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b2">I</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> </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">I</font> being  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <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> <font color="Olive" title="b1">I</font> holds <br/><span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b1">I</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> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">I</font> be   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <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> <font color="Maroon" title="c3">I</font> implies <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">I</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>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:99"/><span class="lab"><font color="Green" title="E73">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">I</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> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c1">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> ;<br/>
<a NAME="E3:99"/><span class="lab"><font color="Green" title="E74">A2</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T65" target="_self" title="RLAFFIN1:th.65">Th65</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">v1</font>, <font color="Maroon" title="c6">v2</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"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c5">v1</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">I</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> &amp; <font color="Maroon" title="c6">v2</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">I</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> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</font></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> <font color="Olive" title="b1">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span> <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">I</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><br/><span class="kw">let </span><font color="Maroon" title="c7">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c5">v1</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">I</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> &amp; <font color="Maroon" title="c6">v2</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">I</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> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</font></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> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span> <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">I</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><br/><span class="kw">set </span><font color="Maroon" title="c8">rv12</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</font></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> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:99_1"/><span class="lab"><font color="Green" title="E75">A3</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">and </span><br/><a NAME="E2:99_1"/><span class="lab"><font color="Green" title="E76">A4</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <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="c5">v1</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">I</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> &amp; <font color="Maroon" title="c6">v2</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">I</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> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</font></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> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span> <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">I</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><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:99_1"/><span class="lab"><font color="Green" title="E77">A5</font></span>: 
<font color="Maroon" title="c5">v1</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">I</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">and </span><br/><a NAME="E4:99_1"/><span class="lab"><font color="Green" title="E78">A6</font></span>: 
<font color="Maroon" title="c6">v2</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">I</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</font></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> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span> <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">I</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><br/><a NAME="E5:99_1"/><span class="lab"><font color="Green" title="E79">A7</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_1"><span class="lab"><font color="Green" title="E76">A4</font></span></a>, <a class="ref" href="xreal_1.html#T12" title="XREAL_1:th.12">XREAL_1:12</a></span>;<br/><a NAME="E6:99_1"/><span class="lab"><font color="Green" title="E80">A8</font></span>: 
<font color="Maroon" title="c6">v2</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">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_1"><span class="lab"><font color="Green" title="E78">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/><a NAME="E7:99_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T71" target="_self" title="RLAFFIN1:th.71">Th71</a></span>;<br/><a NAME="E8:99_1"/><span class="lab"><font color="Green" title="E82">A10</font></span>: 
<font color="Maroon" title="c5">v1</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">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_1"><span class="lab"><font color="Green" title="E77">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/><a NAME="E9:99_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T71" target="_self" title="RLAFFIN1:th.71">Th71</a></span>;<br/><a NAME="E10:99_1"/><span class="lab"><font color="Green" title="E84">A12</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</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> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E8:99_1"><span class="lab"><font color="Green" title="E82">A10</font></span></a>, <a class="txt" href="#E6:99_1"><span class="lab"><font color="Green" title="E80">A8</font></span></a>, <a class="ref" href="rlaffin1.html#T70" target="_self" title="RLAFFIN1:th.70">Th70</a></span>;<br/><div><span class="lab"><font color="Green" title="E85">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">w</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="c9">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</font></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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">w</font> )</span><br/><span class="kw">assume </span><a NAME="E1:99_1_1"/>
<font color="Maroon" title="c9">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</font></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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">w</font></span><br/><span class="lab"><font color="Green" title="E86">A14</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</font></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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">w</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">w</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="c7">r</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:99_1"><span class="lab"><font color="Green" title="E84">A12</font></span></a>, <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"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">w</font></span>)</span></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="c7">r</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D13" title="RLVECT_2:def.13">RLVECT_2:def 13</a></span>
<br/>.= 
<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> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">w</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D13" title="RLVECT_2:def.13">RLVECT_2:def 13</a></span>
;<br/><a NAME="E3:99_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">w</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>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">w</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="#E8:99_1"><span class="lab"><font color="Green" title="E82">A10</font></span></a>, <a class="txt" href="#E6:99_1"><span class="lab"><font color="Green" title="E80">A8</font></span></a>, <a class="ref" href="rlaffin1.html#T71" target="_self" title="RLAFFIN1:th.71">Th71</a></span>;<br/><span class="kw">hence </span><a NAME="E4:99_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</font></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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_1"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="txt" href="#E5:99_1"><span class="lab"><font color="Green" title="E79">A7</font></span></a>, <a class="txt" href="#E2:99_1_1"><span class="lab"><font color="Green" title="E86">A14</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><a NAME="E12:99_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</font></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> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E8:99_1"><span class="lab"><font color="Green" title="E82">A10</font></span></a>, <a class="txt" href="#E6:99_1"><span class="lab"><font color="Green" title="E80">A8</font></span></a>, <a class="ref" href="rusub_4.html#D5" title="RUSUB_4:def.5">RUSUB_4:def 5</a></span>;<br/><a NAME="E13:99_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</font></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> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span> <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">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:99_1"><span class="lab"><font color="Green" title="E85">A13</font></span></a>, <a class="ref" href="rlaffin1.html#T73" target="_self" title="RLAFFIN1:th.73">Th73</a></span>;<br/><a NAME="E14:99_1"/>
<font color="Maroon" title="c5">v1</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="#E3:99_1"><span class="lab"><font color="Green" title="E77">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/><a NAME="E15:99_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:99_1"><span class="lab"><font color="Green" title="E82">A10</font></span></a>, <a class="ref" href="rlaffin1.html#T72" target="_self" title="RLAFFIN1:th.72">Th72</a></span>;<br/><a NAME="E16:99_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_1"><span class="lab"><font color="Green" title="E83">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E17:99_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A16</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_1"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E18:99_1"/>
<font color="Maroon" title="c6">v2</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="#E4:99_1"><span class="lab"><font color="Green" title="E78">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/><a NAME="E19:99_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_1"><span class="lab"><font color="Green" title="E80">A8</font></span></a>, <a class="ref" href="rlaffin1.html#T72" target="_self" title="RLAFFIN1:th.72">Th72</a></span>;<br/><a NAME="E20:99_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99_1"><span class="lab"><font color="Green" title="E81">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E21:99_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:99_1"><span class="lab"><font color="Green" title="E79">A7</font></span></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/><a NAME="E22:99_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A17</font></span>: 
<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> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99_1"><span class="lab"><font color="Green" title="E87">A16</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</font></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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</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="c7">r</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:99_1"><span class="lab"><font color="Green" title="E84">A12</font></span></a>, <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"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font></span>)</span></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="c7">r</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D13" title="RLVECT_2:def.13">RLVECT_2:def 13</a></span>
<br/>.= 
<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> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D13" title="RLVECT_2:def.13">RLVECT_2:def 13</a></span>
;<br/><a NAME="E24:99_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</font></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> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E13:99_1"><span class="lab"><font color="Green" title="E86">A15</font></span></a>, <a class="txt" href="#E22:99_1"><span class="lab"><font color="Green" title="E88">A17</font></span></a>, <a class="ref" href="rlaffin1.html#T72" target="_self" title="RLAFFIN1:th.72">Th72</a></span>;<br/><span class="kw">hence </span><a NAME="E25:99_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v1</font></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> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v2</font></span>)</span> <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">I</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="#E13:99_1"><span class="lab"><font color="Green" title="E86">A15</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:99"/>
<span class="p1">(<span class="default"><a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">I</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#D2" title="CONVEX1:def.2">CONVEX1:def 2</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>
