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<span class="kw">let </span><font color="Maroon" title="c1">M1</font>, <font color="Maroon" title="c2">M2</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" 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">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M1</font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c2">M2</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  iff  for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b3">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M2</font><br/>  for <font color="Olive" title="b4">r</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> holds <br/> ex <font color="Olive" title="b5">s</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <font color="Olive" title="b1">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">p</font>,<font color="Olive" title="b5">s</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b3">q</font>,<font color="Olive" title="b4">r</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M1</font></span>)</span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c2">M2</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="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  iff  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M2</font><br/>  for <font color="Olive" title="b3">r</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> holds <br/> ex <font color="Olive" title="b4">s</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">s</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font>) )</span><br/>





<div><span class="kw">hereby </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">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M2</font><br/>  for <font color="Olive" title="b3">r</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> holds <br/> ex <font color="Olive" title="b4">s</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">s</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font>) ) implies <font color="Maroon" title="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:17_1"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</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">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M2</font><br/>  for <font color="Olive" title="b3">r</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> holds <br/> ex <font color="Olive" title="b4">s</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">s</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M1</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">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M2</font><br/>  for <font color="Olive" title="b2">r</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font> holds <br/> ex <font color="Olive" title="b3">s</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c4">p</font>,<font color="Olive" title="b3">s</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M2</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">r</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font> holds <br/> ex <font color="Olive" title="b2">s</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c4">p</font>,<font color="Olive" title="b2">s</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">q</font>,<font color="Olive" title="b1">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">r</font> be   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <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"> ( <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font> implies  ex <font color="Olive" title="b1">s</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">r</font>) )</span><br/><span class="kw">assume </span><a NAME="E2:17_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</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">  ex <font color="Olive" title="b1">s</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">r</font>)</span><br/><span class="kw">consider </span><font color="Maroon" title="c7">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:17_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c7">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E5:17_1"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
 for <font color="Olive" title="b1">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b2">w1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M2</font>  st <font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> &amp;  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">w</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">s</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">q</font>,<font color="Olive" title="b2">w1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="uniform1.html#T4" onmouseover="rs('uniform1/T4')" onmouseout="rh()">UNIFORM1:4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c7">s</font> as   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c9">s</font> = <font color="Maroon" title="c8">s</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">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">s</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">r</font>)</span><br/><span class="kw">thus </span><a NAME="E7:17_1"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">s</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">y</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="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">s</font>)</span>)</span> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">r</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">s</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="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">r</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:17_1_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">s</font>)
 <span class="kw">and </span><br/><a NAME="E4:17_1_1"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c11">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M1</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">y1</font> = <font color="Maroon" title="c10">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a></span>;<br/>
<a NAME="E7:17_1_1"/>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c12">x1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="metric_1.html#T11" onmouseover="rs('metric_1/T11')" onmouseout="rh()">METRIC_1:11</a></span>;<br/>
<a NAME="E8:17_1_1"/><span class="kw">then </span>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">y1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E5:17_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:17_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T11" onmouseover="rs('metric_1/T11')" onmouseout="rh()">METRIC_1:11</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></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:17"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M2</font><br/>  for <font color="Olive" title="b3">r</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> holds <br/> ex <font color="Olive" title="b4">s</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">s</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">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">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/>

<a NAME="E3:17"/>
 for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b3">u1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M2</font>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b3">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">u</font> holds <br/> ex <font color="Olive" title="b4">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b4">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b5">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b6">w1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M2</font>  st <font color="Olive" title="b6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">w</font> &amp;  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Olive" title="b2">u</font>,<font color="Olive" title="b5">w</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">s</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Olive" title="b3">u1</font>,<font color="Olive" title="b6">w1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">r</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">  for <font color="Olive" title="b1">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b2">u1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M2</font>  st <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b2">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> holds <br/> ex <font color="Olive" title="b3">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b3">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b4">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b5">w1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M2</font>  st <font color="Olive" title="b5">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">w</font> &amp;  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Olive" title="b1">u</font>,<font color="Olive" title="b4">w</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">s</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Olive" title="b2">u1</font>,<font color="Olive" title="b5">w1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M1</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">u1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M2</font>  st <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b1">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u</font> holds <br/> ex <font color="Olive" title="b2">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b3">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b4">w1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M2</font>  st <font color="Olive" title="b4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">w</font> &amp;  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">u</font>,<font color="Olive" title="b3">w</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Olive" title="b1">u1</font>,<font color="Olive" title="b4">w1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">u1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M2</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="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u</font> implies  ex <font color="Olive" title="b1">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b2">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b3">w1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M2</font>  st <font color="Olive" title="b3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">w</font> &amp;  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">u</font>,<font color="Olive" title="b2">w</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c6">u1</font>,<font color="Olive" title="b3">w1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:17_2"/>
( <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u</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">  ex <font color="Olive" title="b1">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b2">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b3">w1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M2</font>  st <font color="Olive" title="b3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">w</font> &amp;  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">u</font>,<font color="Olive" title="b2">w</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c6">u1</font>,<font color="Olive" title="b3">w1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">s</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:17_2"/><span class="lab"><font color="Green" title="E8">A8</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">s</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c4">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E7">A7</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">s</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">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b1">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b2">w1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M2</font>  st <font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> &amp;  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">u</font>,<font color="Olive" title="b1">w</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">s</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c6">u1</font>,<font color="Olive" title="b2">w1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:17_2"/>
<font color="Maroon" title="c7">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M1</font><br/>  for <font color="Olive" title="b2">w1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M2</font>  st <font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> &amp;  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">u</font>,<font color="Olive" title="b1">w</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">s</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c6">u1</font>,<font color="Olive" title="b2">w1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M1</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">w1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M2</font>  st <font color="Olive" title="b1">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">w</font> &amp;  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c8">w</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">s</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c6">u1</font>,<font color="Olive" title="b1">w1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">w1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M2</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="c9">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">w</font> &amp;  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c8">w</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">s</font> implies  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c9">w1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> )</span><br/>



<span class="kw">assume </span><a NAME="E5:17_2"/><span class="lab"><font color="Green" title="E9">A9</font></span>: 
<font color="Maroon" title="c9">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">w</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"> (  not  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c8">w</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">s</font> or  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c9">w1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E6:17_2"/>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c8">w</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">s</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">u1</font>,<font color="Maroon" title="c9">w1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font></span><br/>

<a NAME="E7:17_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">s</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T11" onmouseover="rs('metric_1/T11')" onmouseout="rh()">METRIC_1:11</a></span>;<br/>
<a NAME="E8:17_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_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="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c7">s</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:17_2"/>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c9">w1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_2"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="txt" href="#E5:17_2"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="ref" href="metric_1.html#T11" onmouseover="rs('metric_1/T11')" onmouseout="rh()">METRIC_1:11</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="E4:17"/>
<font color="Maroon" title="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uniform1.html#T3" onmouseover="rs('uniform1/T3')" onmouseout="rh()">UNIFORM1:3</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>
