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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rltopsp1.html#NM1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="convex1.html#V1">convex</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:116_1"/><i><font color="Green">E74</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:116_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><a NAME="E3:116_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:116_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="rltopsp1.html#E2:116_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="convex1.html#T4">CONVEX1:4</a></i>;<br/><a NAME="E4:116_1"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/><a NAME="E5:116_1"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T63" target="_self">Th63</a></i>;<br/><a NAME="E6:116_1"/><i><font color="Green">E78</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:116_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="rltopsp1.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E7:116_1"/>
0 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i><a class="txt" href="rltopsp1.html#E2:116_1"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E8:116_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E9:116_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T53" target="_self">Th53</a></i>;<br/><b>hence </b><a NAME="E10:116_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E4:116_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="rltopsp1.html#E5:116_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="rltopsp1.html#E6:116_1"><i><font color="Green">E78</font></i></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="E2:116"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></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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