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<span class="kw">let </span><font color="Maroon" title="c4">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   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="c4">V</font>  st <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="b2">L</font></span>)</span> where <font color="Olive" title="b2">L</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Olive" title="b1">M</font> : <font color="Olive" title="b2">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c4">V</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">M</font> holds <br/><font color="Olive" title="b1">M</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">M</font> be   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="c4">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c5">M</font> : <font color="Olive" title="b1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c4">V</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">M</font> implies <font color="Maroon" title="c5">M</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:6"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<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="c5">M</font> : <font color="Olive" title="b1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c4">V</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">M</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">M</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> </span><br/>

<a NAME="E2:6"/>
 for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font><br/>  for <font color="Olive" title="b3">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">u</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c6">S</font> = <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="c5">M</font> : <font color="Olive" title="b1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c4">V</font> </span> } </span> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">u</font>, <font color="Maroon" title="c8">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">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="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Maroon" title="c8">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> 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="c7">u</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="c8">v</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font> &amp; <font color="Maroon" title="c9">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Maroon" title="c8">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:6_1"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font> &amp; <font color="Maroon" title="c9">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:6_1"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 <span class="kw">and </span><br/><a NAME="E3:6_1"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c8">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">Lv</font> being   <a href="convex2.html#NM1" title="CONVEX2:NM.1">Convex_Combination</a> of <font color="Maroon" title="c4">V</font><span class="kw"> such that </span><br/><a NAME="E5:6_1"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c10">Lv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font>
 <span class="kw">and </span><br/><a NAME="E6:6_1"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
 for <font color="Olive" title="b1">A</font> being   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="c4">V</font>  st <font color="Maroon" title="c8">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon" title="c10">Lv</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Olive" title="b1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="convex3.html#T1" target="_self" onmouseover="rs('convex3/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">Lv</font> = <font color="Maroon" title="c10">Lv</font> as   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c5">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E6:6_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">Lu</font> being   <a href="convex2.html#NM1" title="CONVEX2:NM.1">Convex_Combination</a> of <font color="Maroon" title="c4">V</font><span class="kw"> such that </span><br/><a NAME="E9:6_1"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c12">Lu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u</font>
 <span class="kw">and </span><br/><a NAME="E10:6_1"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
 for <font color="Olive" title="b1">A</font> being   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="c4">V</font>  st <font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon" title="c12">Lu</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Olive" title="b1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="convex3.html#T1" target="_self" onmouseover="rs('convex3/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">Lu</font> = <font color="Maroon" title="c12">Lu</font> as   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c5">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E10:6_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E12">A9</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v</font></span>)</span> = 
<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="c9">r</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c13">Lu</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></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="c11">Lv</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:6_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E5:6_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="rlvect_3.html#T2" onmouseover="rs('rlvect_3/T2')" onmouseout="rh()">RLVECT_3:2</a></span>
<br/>.= 
<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="c9">r</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c13">Lu</font></span>)</span></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"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c11">Lv</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_3.html#T2" onmouseover="rs('rlvect_3/T2')" onmouseout="rh()">RLVECT_3:2</a></span>
<br/>.= 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c13">Lu</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c11">Lv</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_3.html#T1" onmouseover="rs('rlvect_3/T1')" onmouseout="rh()">RLVECT_3:1</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c9">r</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E14:6_1"/><span class="lab"><font color="Green" title="E13">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c13">Lu</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r</font></span>)</span> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c11">Lv</font></span>)</span> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c5">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="convex2.html#T9" onmouseover="rs('convex2/T9')" onmouseout="rh()">CONVEX2:9</a></span>;<br/>
<a NAME="E15:6_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c13">Lu</font></span>)</span> <a href="rlvect_2.html#K7" title="RLVECT_2:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r</font></span>)</span> <a href="rlvect_2.html#K8" title="RLVECT_2:func.8">*</a> <font color="Maroon" title="c11">Lv</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c4">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex3.html#D1" target="_self" onmouseover="rs('convex3/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E16:6_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="c5">M</font> : <font color="Olive" title="b1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c4">V</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:6_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E14:6_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:6_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:6"/>
<font color="Maroon" title="c5">M</font> 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" onmouseover="rs('convex1/D2')" onmouseout="rh()">CONVEX1:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
