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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/> ( <span class="p1">(<span class="default"><a href="metric_3.html#K1" title="METRIC_3:func.1">dist_cart2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="metric_3.html#K1" title="METRIC_3:func.1">dist_cart2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c4">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">x2</font> = <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> , <font color="Maroon" title="c8">y2</font> = <font color="Maroon" title="c4">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></span>;<br/>
<a NAME="E3:7"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c8">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T24" title="MCART_1:th.24">MCART_1:24</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:7"/>
( <span class="p1">(<span class="default"><a href="metric_3.html#K1" title="METRIC_3:func.1">dist_cart2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> implies <span class="p1">(<span class="default"><a href="metric_3.html#K1" title="METRIC_3:func.1">dist_cart2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c9">d1</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">d2</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">y2</font>;<br/>
<span class="kw">assume </span><a NAME="E1:7_1"/>
<span class="p1">(<span class="default"><a href="metric_3.html#K1" title="METRIC_3:func.1">dist_cart2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font></span><br/>

<a NAME="E2:7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="metric_3.html#D1" target="_self" title="METRIC_3:def.1">Def1</a></span>;<br/>
<a NAME="E3:7_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">y2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></span>;<br/>
<a NAME="E4:7_1"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/>
<a NAME="E5:7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a></span>;<br/>
<a NAME="E6:7_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E3:7_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:7_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E5:7_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E5:7"/><span class="lab"><font color="Green" title="E4">A5</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="metric_3.html#K1" title="METRIC_3:func.1">dist_cart2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E6:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A6</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/>
<span class="p1">(<span class="default"><a href="metric_3.html#K1" title="METRIC_3:func.1">dist_cart2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">y2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="metric_3.html#D1" target="_self" title="METRIC_3:def.1">Def1</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E4">A5</font></span></a>, <a class="txt" href="#E6:7"><span class="lab"><font color="Green" title="E5">A6</font></span></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:7"/>
<span class="p1">(<span class="default"><a href="metric_3.html#K1" title="METRIC_3:func.1">dist_cart2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
