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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



<a NAME="E1:2"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_5.html#D2" onmouseover="rs('rusub_5/D2')" onmouseout="rh()">RUSUB_5:def 2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rltopsp1.html#D6" onmouseover="rs('rltopsp1/D6')" onmouseout="rh()">RLTOPSP1:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c4">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <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/>

<span class="kw">assume </span><a NAME="E2:2"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><font color="Maroon" title="c4">r</font></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A2</font></span>: 
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#D6" onmouseover="rs('rltopsp1/D6')" onmouseout="rh()">RLTOPSP1:def 6</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E4:2"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<a NAME="E5:2"/>
<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="tarski.html#R2" title="TARSKI: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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#D1" onmouseover="rs('convex1/D1')" onmouseout="rh()">CONVEX1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x9</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E7:2"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<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">x9</font>
 <span class="kw">and </span><br/><a NAME="E8:2"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c6">x9</font> <a href="tarski.html#R2" title="TARSKI: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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2"><span class="lab"><font color="Green" title="E5">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E10:2"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<font color="Maroon" title="c6">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">v</font>
 <span class="kw">and </span><br/><a NAME="E11:2"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
( <font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E8:2"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c4">r</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E13:2"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
( <font color="Maroon" title="c9">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="rltopsp1.html#T1" onmouseover="rs('rltopsp1/T1')" onmouseout="rh()">RLTOPSP1:1</a></span>;<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="c9">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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E10:2"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="rlvect_1.html#T34" onmouseover="rs('rlvect_1/T34')" onmouseout="rh()">RLVECT_1:34</a></span>;<br/>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E13:2"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="circled1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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