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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</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">A</font>, <font color="Olive" title="b2">B</font> being <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Olive" title="b1">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Olive" title="b2">B</font> is <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a></span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c3">B</font> is <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a></span><br/><b>let </b><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D6" title="RLTOPSP1:def.6">RLTOPSP1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 or <font color="Maroon" title="c4">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c3">B</font> )</span><br/>





<b>assume </b><a NAME="E1:2"/><i><font color="Green" title="E3">A1</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c3">B</font></span><br/>

<a NAME="E2:2"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> where <font color="Olive" title="b1">v</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c3">B</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="convex1.html#D1" title="CONVEX1:def.1">CONVEX1:def 1</a></i>;<br/>
<a NAME="E3:2"/><i><font color="Green" title="E5">A3</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b2">v</font></span>)</span> where <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_5.html#D2" title="RUSUB_5:def.2">RUSUB_5:def 2</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c3">B</font></span>)</span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c3">B</font> )</span><br/>

<b>assume </b><a NAME="E4:2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c3">B</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c3">B</font></span><br/>

<b>then consider </b><font color="Maroon" title="c6">x'</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E6:2"/><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x'</font>
 <b>and </b><br/><a NAME="E7:2"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c6">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="#E2:2"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">u</font>, <font color="Maroon" title="c8">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E9:2"/><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon" title="c6">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c8">v</font>
 <b>and </b><br/><a NAME="E10:2"/><i><font color="Green" title="E9">A7</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>and </b><br/><a NAME="E11:2"/><i><font color="Green" title="E10">A8</font></i>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="#E3:2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="#E7:2"><i><font color="Green" title="E7">A5</font></i></a></i>;<br/>
<a NAME="E12:2"/><i><font color="Green" title="E11">A9</font></i>: 
( <font color="Maroon" title="c4">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="txt" href="#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="rltopsp1.html#D6" title="RLTOPSP1:def.6">RLTOPSP1:def 6</a></i>;<br/>
<a NAME="E13:2"/><i><font color="Green" title="E12">A10</font></i>: 
( <font color="Maroon" title="c4">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="txt" href="#E10:2"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="#E11:2"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="rltopsp1.html#T1" title="RLTOPSP1:th.1">RLTOPSP1:1</a></i>;<br/>
<a NAME="E14:2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="#E9:2"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="rlvect_1.html#T48" title="RLVECT_1:th.48">RLVECT_1:48</a></i>;<br/>
<b>hence </b><a NAME="E15:2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="#E12:2"><i><font color="Green" title="E11">A9</font></i></a>, <a class="txt" href="#E13:2"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="circled1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></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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