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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">A</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b2">B</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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><br/>  for <font color="Olive" title="b3">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<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="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Olive" title="b2">B</font></span>)</span>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b3">C</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  holds <br/><font color="Olive" title="b1">A</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</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#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">  for <font color="Olive" title="b1">B</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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><br/>  for <font color="Olive" title="b2">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<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="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Olive" title="b1">B</font></span>)</span>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b2">C</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  holds <br/><font color="Maroon" title="c2">A</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">B</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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">  for <font color="Olive" title="b1">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<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="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">B</font></span>)</span>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b1">C</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  holds <br/><font color="Maroon" title="c2">A</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<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="topmetr.html#K1" title="TOPMETR: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="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c4">C</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  implies <font color="Maroon" title="c2">A</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:70"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E2:70"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
<font color="Maroon" title="c4">C</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</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"> <font color="Maroon" title="c2">A</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">r0</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:70"/><span class="lab"><font color="Green" title="E50">A3</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="c5">r0</font>
 <span class="kw">and </span><br/><a NAME="E5:70"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<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="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">B</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">C</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E49">A2</font></span></a></span>;<br/>
<a NAME="E6:70"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> implies  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_1"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</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">  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r0</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x0</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">y0</font> = <font color="Maroon" title="c7">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<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="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">B</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E48">A1</font></span></a></span>;<br/>
<a NAME="E3:70_1"/>
(  the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <span class="p1">(<span class="default"><span class="p2">(<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="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c8">x0</font>,<font color="Maroon" title="c9">y0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</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="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>) &amp;  the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <span class="p1">(<span class="default"><span class="p2">(<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="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c8">x0</font>,<font color="Maroon" title="c9">y0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c8">x0</font>,<font color="Maroon" title="c9">y0</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#D1" onmouseover="rs('topmetr/D1')" onmouseout="rh()">TOPMETR:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:70_1"/>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E5:70"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E1:70_1"><span class="lab"><font color="Green" title="E52">A5</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:70"/>
<font color="Maroon" title="c2">A</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E50">A3</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/>


</div>
