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<b>let </b><font color="Maroon">X</font> be   <a href="rltopsp1.html#NM1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon">C</font> be  <a href="convex1.html#V1">convex</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:118_1"/><i><font color="Green">E36</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E2:118_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><a NAME="E3:118_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">C</font></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="convex1.html#T4">CONVEX1:4</a></i>;<br/><a NAME="E4:118_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">C</font></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/><a NAME="E5:118_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">C</font></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E6:118_1"/><i><font color="Green">E49</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E7:118_1"/>
0 <a href="real_1.html#K3">+</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i/>;<br/><a NAME="E8:118_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E9:118_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:118_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="rltopsp1.html#T12" target="_self">Th12</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:118"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">C</font> is <a href="convex1.html#V1">convex</a>
 <b>by </b><i><a class="ref" href="convex1.html#T4">CONVEX1:4</a></i>;<br/>


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