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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="rltopsp1.html#NM1" title="RLTOPSP1:NM.1">LinearTopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">r</font> being non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Olive" title="b2">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b1">V</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Olive" title="b1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">r</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font></span>)</span></span><br/>





<a NAME="E1:101"/>
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="ref" href="convex1.html#T39" title="CONVEX1:th.39">CONVEX1:39</a>, <a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<b>hence </b><a NAME="E2:101"/>
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T50" target="_self" title="RLTOPSP1:th.50">Th50</a>, <a class="ref" href="tops_1.html#T56" title="TOPS_1:th.56">TOPS_1:56</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">V</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">V</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E3:101"/><i><font color="Green" title="E63">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">V</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>consider </b><font color="Maroon" title="c6">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:101"/><i><font color="Green" title="E64">A2</font></i>: 
<font color="Maroon" title="c6">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E7:101"/><i><font color="Green" title="E65">A3</font></i>: 
<font color="Maroon" title="c6">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font>
 <b>and </b><br/><a NAME="E8:101"/><i><font color="Green" title="E66">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E3:101"><i><font color="Green" title="E63">A1</font></i></a>, <a class="ref" href="tops_1.html#T54" title="TOPS_1:th.54">TOPS_1:54</a></i>;<br/>
<a NAME="E9:101"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E7:101"><i><font color="Green" title="E65">A3</font></i></a>, <a class="ref" href="convex1.html#T39" title="CONVEX1:th.39">CONVEX1:39</a></i>;<br/>
<a NAME="E10:101"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="ref" href="convex1.html#T37" title="CONVEX1:th.37">CONVEX1:37</a></i>;<br/>
<a NAME="E11:101"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 1 <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/>
<a NAME="E12:101"/><b>then </b><i><font color="Green" title="E67">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="ref" href="convex1.html#T32" title="CONVEX1:th.32">CONVEX1:32</a></i>;<br/>
<a NAME="E13:101"/><i><font color="Green" title="E68">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">Q</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E8:101"><i><font color="Green" title="E66">A4</font></i></a></i>;<br/>
<a NAME="E14:101"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E6:101"><i><font color="Green" title="E64">A2</font></i></a>, <a class="ref" href="rltopsp1.html#T50" target="_self" title="RLTOPSP1:th.50">Th50</a></i>;<br/>
<a NAME="E15:101"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E12:101"><i><font color="Green" title="E67">A5</font></i></a>, <a class="txt" href="rltopsp1.html#E13:101"><i><font color="Green" title="E68">A6</font></i></a>, <a class="ref" href="tops_1.html#T54" title="TOPS_1:th.54">TOPS_1:54</a></i>;<br/>
<a NAME="E16:101"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">V</font></span>)</span>
 
;<br/>
<a NAME="E17:101"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/>
<a NAME="E18:101"/><b>then </b>
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E19:101"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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