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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">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">r</font> be  non <a href="xboole_0.html#V1">zero</a> <a href="real_1.html#NM1">Real</a>;<br/>





<a NAME="E1:101"/><i><font color="Green">E36</font></i>: 
<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">V</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:101"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E3:101"/><b>then </b>
<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">V</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="convex1.html#T39">CONVEX1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:101"/>
<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">V</font></span>)</span> <a href="tarski.html#R1">c=</a>  <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">V</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:101"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <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">V</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/>
<b>consider </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:101"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E9:101"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E10:101"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T54">TOPS_1:54</a></i>;<br/>
<a NAME="E11:101"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="convex1.html#T39">CONVEX1:39</a></i>;<br/>
<a NAME="E12:101"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="convex1.html#T37">CONVEX1:37</a></i>;<br/>
<a NAME="E13:101"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> 1 <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/>
<a NAME="E14:101"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="convex1.html#T32">CONVEX1:32</a></i>;<br/>
<a NAME="E15:101"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">Q</font>
 
<b>by </b><i/>;<br/>
<a NAME="E16:101"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:101"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="rltopsp1.html#T8" target="_self">Th8</a>, <a class="ref" href="tops_1.html#T54">TOPS_1:54</a></i>;<br/>
<a NAME="E18:101"/><b>then </b>
<font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</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">V</font></span>)</span>
 
;<br/>
<a NAME="E19:101"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</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">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>
<a NAME="E20:101"/><b>then </b>
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</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">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E21:101"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>


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