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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">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</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#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b4">u</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="b2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</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="Olive" title="b4">u</font>,<font color="Olive" title="b1">r</font>) &amp; <font color="Olive" title="b3">q</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="Olive" title="b4">u</font>,<font color="Olive" title="b1">r</font>) holds <br/> ex <font color="Olive" title="b5">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b5">h</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Olive" title="b5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b5">h</font> <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="b4">u</font>,<font color="Olive" title="b1">r</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">p</font>, <font color="Olive" title="b2">q</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#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b3">u</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">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</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="Olive" title="b3">u</font>,<font color="Maroon" title="c2">r</font>) &amp; <font color="Olive" title="b2">q</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="Olive" title="b3">u</font>,<font color="Maroon" title="c2">r</font>) holds <br/> ex <font color="Olive" title="b4">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b4">h</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Olive" title="b4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b4">h</font> <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">u</font>,<font color="Maroon" title="c2">r</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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#K15" title="EUCLID:func.15">TOP-REAL</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">u</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="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c3">p</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="Olive" title="b1">u</font>,<font color="Maroon" title="c2">r</font>) &amp; <font color="Maroon" title="c4">q</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="Olive" title="b1">u</font>,<font color="Maroon" title="c2">r</font>) holds <br/> ex <font color="Olive" title="b2">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b2">h</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">h</font> <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">u</font>,<font color="Maroon" title="c2">r</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">u</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="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c3">p</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="c2">r</font>) &amp; <font color="Maroon" title="c4">q</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="c2">r</font>) implies  ex <font color="Olive" title="b1">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">h</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">h</font> <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">u</font>,<font color="Maroon" title="c2">r</font>) ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:79"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">and </span><br/><a NAME="E2:79"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
( <font color="Maroon" title="c3">p</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="c2">r</font>) &amp; <font color="Maroon" title="c4">q</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="c2">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">  ex <font color="Olive" title="b1">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">h</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">h</font> <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">u</font>,<font color="Maroon" title="c2">r</font>) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">Q</font> =  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c2">r</font>) as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T8" onmouseover="rs('topreal3/T8')" onmouseout="rh()">TOPREAL3:8</a></span>;<br/>
<a NAME="E4:79"/>
<font color="Maroon" title="c6">Q</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T70" target="_self" onmouseover="rs('jordan2c/T70')" onmouseout="rh()">Th55</a></span>;<br/>
<a NAME="E5:79"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A3</font></span>: 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>) <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">u</font>,<font color="Maroon" title="c2">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="jordan1.html#D1" onmouseover="rs('jordan1/D1')" onmouseout="rh()">JORDAN1:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">P</font> =  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>) as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<a NAME="E7:79"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="topreal1.html#T9" onmouseover="rs('topreal1/T9')" onmouseout="rh()">TOPREAL1:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">P</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E9:79"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
<font color="Maroon" title="c8">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E10:79"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
( <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D1" onmouseover="rs('topreal1/D1')" onmouseout="rh()">TOPREAL1:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">h</font> = <font color="Maroon" title="c8">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T3" onmouseover="rs('jordan6/T3')" onmouseout="rh()">JORDAN6:3</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">h</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">h</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">h</font> <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">u</font>,<font color="Maroon" title="c2">r</font>) )</span><br/>

<a NAME="E12:79"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">P</font></span>)</span> &amp; <font color="Maroon" title="c8">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="txt" href="#E9:79"><span class="lab"><font color="Green" title="E58">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:79"/>
( <font color="Maroon" title="c9">h</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">h</font> <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">u</font>,<font color="Maroon" title="c2">r</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="txt" href="#E10:79"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="jordan6.html#T3" onmouseover="rs('jordan6/T3')" onmouseout="rh()">JORDAN6:3</a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</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/>


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