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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">z</font>, <font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>







<b>thus </b><a NAME="E1:54"/>
( <font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K5" title="BHSP_2:func.5">cl_Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font> implies  <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1"/>
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K5" title="BHSP_2:func.5">cl_Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>
 ;<br/>

<a NAME="E2:54_1"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">z</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="bhsp_2.html#T47" target="_self" title="BHSP_2:th.47">Th47</a></i>;<br/>
<b>hence </b><a NAME="E3:54_1"/>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D5" title="BHSP_1:def.5">BHSP_1:def 5</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:54"/>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font>
 ;<br/>

<a NAME="E3:54"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">z</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#D5" title="BHSP_1:def.5">BHSP_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:54"/>
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K5" title="BHSP_2:func.5">cl_Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>
 ;<br/>


</div>
