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<div class="add">

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  #);<br/>
<a NAME="E1:30_1_1"/>
 <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  #) is   <a href="sub_metr.html#NM1">PseudoMetricSpace</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  #);<br/><a NAME="E1:30_1_1_1_1"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><b>hence </b><a NAME="E2:30_1_1_1_1"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="sub_metr.html#T40" target="_self">Th40</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  #);<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E1:30_1_1_1_2"/><i><font color="Green">E19</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E2:30_1_1_1_2"/><i><font color="Green">E20</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E3:30_1_1_1_2"/><i><font color="Green">E21</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E4:30_1_1_1_2"/><i><font color="Green">E22</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><b>thus </b><a NAME="E5:30_1_1_1_2"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sub_metr.html#E1:30_1_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="sub_metr.html#E2:30_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="sub_metr.html#T42" target="_self">Th42</a></i>;<br/><b>thus </b><a NAME="E6:30_1_1_1_2"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sub_metr.html#E1:30_1_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="sub_metr.html#E3:30_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="sub_metr.html#E4:30_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="sub_metr.html#T43" target="_self">Th43</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:30_1_1_1"/>
 <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<a href="sub_metr.html#K1" target="_self">Set_to_zero</a>  #) is   <a href="sub_metr.html#NM1">PseudoMetricSpace</a>
 <b>by </b><i><a class="txt" href="sub_metr.html#E1:30_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="metric_1.html#T1">METRIC_1:1</a>, <a class="ref" href="metric_1.html#T3">METRIC_1:3</a>, <a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:30_1_1"/>
(  <a href="sub_metr.html#K2" target="_self">ZeroSpace</a>  is <a href="metric_1.html#V6">Reflexive</a> &amp;  <a href="sub_metr.html#K2" target="_self">ZeroSpace</a>  is <a href="metric_1.html#V8">symmetric</a> &amp;  <a href="sub_metr.html#K2" target="_self">ZeroSpace</a>  is <a href="metric_1.html#V9">triangle</a> )
 ;<br/>


</div>
