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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">r</font>, <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">t</font> holds <br/> for <font color="Olive" title="b4">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> holds <br/> for <font color="Olive" title="b5">BA</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b5">BA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">r</font>) holds <br/><font color="Olive" title="b2">s</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b5">BA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b5">BA</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</font>, <font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">t</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t</font> implies  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> holds <br/> for <font color="Olive" title="b2">BA</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b2">BA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">r</font>) holds <br/><font color="Maroon" title="c3">s</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b2">BA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b2">BA</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:13"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">and </span><br/><a NAME="E2:13"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t</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">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> holds <br/> for <font color="Olive" title="b2">BA</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b2">BA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">r</font>) holds <br/><font color="Maroon" title="c3">s</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b2">BA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b2">BA</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</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#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> implies  for <font color="Olive" title="b1">BA</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">BA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">r</font>) holds <br/><font color="Maroon" title="c3">s</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b1">BA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b1">BA</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:13"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</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">  for <font color="Olive" title="b1">BA</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">BA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">r</font>) holds <br/><font color="Maroon" title="c3">s</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b1">BA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b1">BA</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">BA</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</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="c6">BA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">r</font>) implies <font color="Maroon" title="c3">s</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">BA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">BA</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:13"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c6">BA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">r</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="c3">s</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">BA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">BA</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">e</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="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">s</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">BA</font> or <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">BA</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:13"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">s</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">BA</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="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">BA</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:13"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w</font>
 <span class="kw">and </span><br/><a NAME="E8:13"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
<font color="Maroon" title="c8">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">BA</font>
 ;<br/>
<a NAME="E9:13"/>
<font color="Maroon" title="c8">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> :  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">q</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:13"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="metric_1.html#D14" onmouseover="rs('metric_1/D14')" onmouseout="rh()">METRIC_1:def 14</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:13"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font>
 <span class="kw">and </span><br/><a NAME="E12:13"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c9">q</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">p</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w</font>;<br/>
<span class="lab"><font color="Green" title="E15">A9</font></span>: <font color="Maroon" title="c7">e</font> = 
<font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:13"><span class="lab"><font color="Green" title="E11">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">t</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>
<br/>.= 
<font color="Maroon" title="c4">t</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">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T49" onmouseover="rs('rvsum_1/T49')" onmouseout="rh()">RVSUM_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c4">t</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">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w</font></span>)</span>

;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T67" onmouseover="rs('euclid/T67')" onmouseout="rh()">EUCLID:67</a></span>;<br/>
<a NAME="E15:13"/><span class="lab"><font color="Green" title="E16">A10</font></span>: 
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c9">q</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E11:13"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="txt" href="jordan2c.html#E11"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a>, <a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a></span>;<br/>
<a NAME="E16:13"/>
<font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="xreal_1.html#T183" onmouseover="rs('xreal_1/T183')" onmouseout="rh()">XREAL_1:183</a></span>;<br/>
<a NAME="E17:13"/><span class="kw">then </span>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c9">q</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:13"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="metric_1.html#T5" onmouseover="rs('metric_1/T5')" onmouseout="rh()">METRIC_1:5</a>, <a class="ref" href="xreal_1.html#T153" onmouseover="rs('xreal_1/T153')" onmouseout="rh()">XREAL_1:153</a></span>;<br/>
<a NAME="E18:13"/><span class="kw">then </span>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">y</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:13"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E19:13"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> :  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">f</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font> </span> } </span> 
 
;<br/>
<a NAME="E20:13"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">BA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="metric_1.html#D14" onmouseover="rs('metric_1/D14')" onmouseout="rh()">METRIC_1:def 14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:13"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">t</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c6">BA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:13"><span class="lab"><font color="Green" title="E15">A9</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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