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<span class="kw">let </span><font color="Maroon" title="c1">X</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Y</font> being   <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c1">X</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; (  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">X</font> ) holds <br/> ex <font color="Olive" title="b2">n0</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b3">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b4">x0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b4">x0</font>,<font color="Olive" title="b3">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n0</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</font> be   <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>  &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">X</font> ) implies  ex <font color="Olive" title="b1">n0</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b3">x0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b3">x0</font>,<font color="Olive" title="b2">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n0</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1"/><span class="lab"><font color="Green" title="E1">A1</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a> 
 <span class="kw">and </span><br/><a NAME="E2:1"/><span class="lab"><font color="Green" title="E2">A2</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E3:1"/><span class="lab"><font color="Green" title="E3">A3</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n0</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ex <font color="Olive" title="b3">x0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r</font> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b3">x0</font>,<font color="Olive" title="b2">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n0</font> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>,  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>,  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>,  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <span class="kw">means </span>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $3 &amp; $3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> $1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> ($2,$3)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $5 &amp; $5 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> ($4,$5) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> ($2,<span class="p1">(<span class="default">$3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> ($4,$5)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) );<br/>
<span class="kw">assume </span><a NAME="E4:1"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
 for <font color="Olive" title="b1">n0</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b3">x0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  holds <br/>(  not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r</font> or  not  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b3">x0</font>,<font color="Olive" title="b2">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">x0</font> =  the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/><a NAME="E1:1_1"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> ( the   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>,1) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> )
 ;<br/><span class="kw">assume </span><a NAME="E2:1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E3:1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E1:1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a></span>; <a class="txt" 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">then </span><span class="kw">consider </span><font color="Maroon" title="c3">z0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:1"/><span class="lab"><font color="Green" title="E5">A6</font></span>: 
<font color="Maroon" title="c3">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">z0</font> = <font color="Maroon" title="c3">z0</font> 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="txt" href="#E7:1"><span class="lab"><font color="Green" title="E5">A6</font></span></a></span>;<br/>
<a NAME="E9:1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1"><span class="lab"><font color="Green" title="E3">A3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">t01</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:1"/><span class="lab"><font color="Green" title="E6">A7</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">t01</font>
 <span class="kw">and </span><br/><a NAME="E12:1"/><span class="lab"><font color="Green" title="E7">A8</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">z0</font>,<font color="Maroon" title="c5">t01</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1"><span class="lab"><font color="Green" title="E5">A6</font></span></a>, <a class="ref" href="pcomps_1.html#D4" title="PCOMPS_1:def.4">PCOMPS_1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">t0</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<font color="Maroon" title="c5">t01</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E14:1"/>
<font color="Maroon" title="c6">t0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/>
<a NAME="E15:1"/><span class="kw">then </span>
<font color="Maroon" title="c6">t0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E16:1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A9</font></span>: 
<font color="Maroon" title="c6">t0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>;<br/>
<a NAME="E17:1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">z0</font>,<font color="Maroon" title="c6">t0</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">z0</font>,<font color="Maroon" title="c5">t01</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#T1" title="PCOMPS_1:th.1">PCOMPS_1:1</a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/>
<a NAME="E18:1"/><span class="kw">then </span>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">z0</font>,<font color="Maroon" title="c6">t0</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1"><span class="lab"><font color="Green" title="E7">A8</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E19:1"/><span class="kw">then </span>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">z0</font>,<font color="Maroon" title="c6">t0</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T23" title="SUBSET_1:th.23">SUBSET_1:23</a></span>;<br/>
<a NAME="E20:1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A10</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">z0</font>,<font color="Maroon" title="c6">t0</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E21:1"/><span class="lab"><font color="Green" title="E10">A11</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  ex <font color="Olive" title="b4">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b5">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">r</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">r1</font> &amp; <font color="Olive" title="b5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b4">x1</font>,<font color="Olive" title="b5">r1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b2">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b4">x1</font>,<font color="Olive" title="b5">r1</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  ex <font color="Olive" title="b3">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b4">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">r1</font> &amp; <font color="Olive" title="b4">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b3">x1</font>,<font color="Olive" title="b4">r1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b3">x1</font>,<font color="Olive" title="b4">r1</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" 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="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  ex <font color="Olive" title="b2">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b3">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">r</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r1</font> &amp; <font color="Olive" title="b3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b2">x1</font>,<font color="Olive" title="b3">r1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b2">x1</font>,<font color="Olive" title="b3">r1</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b2">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font> &amp; <font color="Maroon" title="c9">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r1</font> &amp; <font color="Olive" title="b2">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">r1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">r1</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )</span><br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:1_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T83" title="NEWTON:th.83">NEWTON:83</a></span>;<br/><a NAME="E2:1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A12</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T139" title="XREAL_1:th.139">XREAL_1:139</a></span>;<br/><a NAME="E3:1_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T83" title="NEWTON:th.83">NEWTON:83</a></span>;<br/><a NAME="E4:1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A13</font></span>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T139" title="XREAL_1:th.139">XREAL_1:139</a>, <a class="ref" href="xreal_1.html#T216" title="XREAL_1:th.216">XREAL_1:216</a></span>;<br/>2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> = 
2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>
;<br/><a NAME="E6:1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A14</font></span>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:1_2_1"/><span class="lab"><font color="Green" title="E14">A15</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font>
 <span class="kw">and </span><br/><a NAME="E8:1_2_1"/>
<font color="Maroon" title="c9">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:1_2_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b2">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>,<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">r1</font>]</span><br/><a NAME="E10:1_2_1"/>
 not  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E7:1_2_1"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="ref" href="xreal_1.html#T215" title="XREAL_1:th.215">XREAL_1:215</a></span>;<br/><a NAME="E11:1_2_1"/><span class="kw">then </span>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T24" title="SUBSET_1:th.24">SUBSET_1:24</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">z0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:1_2_1"/><span class="lab"><font color="Green" title="E15">A16</font></span>: 
<font color="Maroon" title="c10">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c10">z0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_2_1"><span class="lab"><font color="Green" title="E15">A16</font></span></a></span>;<br/><a NAME="E15:1_2_1"/><span class="lab"><font color="Green" title="E16">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1"><span class="lab"><font color="Green" title="E3">A3</font></span></a></span>;<br/><a NAME="E16:1_2_1"/>
(  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="ref" href="pcomps_1.html#T29" title="PCOMPS_1:th.29">PCOMPS_1:29</a></span>;<br/><a NAME="E17:1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#T31" title="PCOMPS_1:th.31">PCOMPS_1:31</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">t02</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E19:1_2_1"/><span class="lab"><font color="Green" title="E17">A18</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">t02</font>
 <span class="kw">and </span><br/><a NAME="E20:1_2_1"/><span class="lab"><font color="Green" title="E18">A19</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">t02</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_2_1"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="ref" href="pcomps_1.html#D4" title="PCOMPS_1:def.4">PCOMPS_1:def 4</a></span>;<br/><a NAME="E21:1_2_1"/><span class="lab"><font color="Green" title="E19">A20</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">t02</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_2_1"><span class="lab"><font color="Green" title="E18">A19</font></span></a>, <a class="ref" href="xboole_1.html#T18" title="XBOOLE_1:th.18">XBOOLE_1:18</a></span>;<br/><a NAME="E22:1_2_1"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_2_1"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">t01</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E24:1_2_1"/><span class="lab"><font color="Green" title="E20">A21</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">t01</font>
 <span class="kw">and </span><br/><a NAME="E25:1_2_1"/><span class="lab"><font color="Green" title="E21">A22</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c13">t01</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:1_2_1"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="ref" href="pcomps_1.html#D4" title="PCOMPS_1:def.4">PCOMPS_1:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">r1</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<font color="Maroon" title="c13">t01</font>,<font color="Maroon" title="c12">t02</font>)</span>)</span>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>) as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E27:1_2_1"/><span class="lab"><font color="Green" title="E22">A23</font></span>: 
<font color="Maroon" title="c14">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<font color="Maroon" title="c13">t01</font>,<font color="Maroon" title="c12">t02</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/><a NAME="E28:1_2_1"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<font color="Maroon" title="c13">t01</font>,<font color="Maroon" title="c12">t02</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">t02</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/><a NAME="E29:1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A24</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c14">r1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">t02</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_2_1"><span class="lab"><font color="Green" title="E22">A23</font></span></a>, <a class="ref" href="pcomps_1.html#T1" title="PCOMPS_1:th.1">PCOMPS_1:1</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E30:1_2_1"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<font color="Maroon" title="c13">t01</font>,<font color="Maroon" title="c12">t02</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">t01</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/><a NAME="E31:1_2_1"/><span class="kw">then </span>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c14">r1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c13">t01</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_2_1"><span class="lab"><font color="Green" title="E22">A23</font></span></a>, <a class="ref" href="pcomps_1.html#T1" title="PCOMPS_1:th.1">PCOMPS_1:1</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E32:1_2_1"/><span class="kw">then </span>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c14">r1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_2_1"><span class="lab"><font color="Green" title="E21">A22</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E33:1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A25</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c14">r1</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T23" title="SUBSET_1:th.23">SUBSET_1:23</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c15">x1</font> = <font color="Maroon" title="c11">x1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c15">x1</font>,<font color="Olive" title="b1">r1</font>]</span><br/><span class="kw">take </span><font color="Maroon" title="c16">r1</font> = <font color="Maroon" title="c14">r1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c16">r1</font>]</span><br/><a NAME="E34:1_2_1"/><span class="lab"><font color="Green" title="E25">A26</font></span>: 
<font color="Maroon" title="c16">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/><a NAME="E35:1_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> (<font color="Maroon" title="c13">t01</font>,<font color="Maroon" title="c12">t02</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_2_1"><span class="lab"><font color="Green" title="E20">A21</font></span></a>, <a class="txt" href="#E19:1_2_1"><span class="lab"><font color="Green" title="E17">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></span>;<br/><span class="kw">hence </span><a NAME="E36:1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c16">r1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_2_1"><span class="lab"><font color="Green" title="E19">A20</font></span></a>, <a class="txt" href="#E6:1_2_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E2:1_2_1"><span class="lab"><font color="Green" title="E11">A12</font></span></a>, <a class="txt" href="#E4:1_2_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="txt" href="#E29:1_2_1"><span class="lab"><font color="Green" title="E23">A24</font></span></a>, <a class="txt" href="#E33:1_2_1"><span class="lab"><font color="Green" title="E24">A25</font></span></a>, <a class="txt" href="#E34:1_2_1"><span class="lab"><font color="Green" title="E25">A26</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></span>; <a class="txt" 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="E2:1_2"/>
 ex <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b2">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font> &amp; <font color="Maroon" title="c9">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">r</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r1</font> &amp; <font color="Olive" title="b2">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">r1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">r1</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 ; <a class="txt" 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>
<a NAME="E22:1"/>
 ex <font color="Olive" title="b1">x0</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b2">r0</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( <font color="Olive" title="b1">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z0</font> &amp; <font color="Olive" title="b2">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t0</font> &amp; (  for <font color="Olive" title="b3">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b2">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">n</font></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <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>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,   <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>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>] <span class="kw">means </span>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $2 <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; $2 <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> $1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default">$2 <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span>,<span class="p2">(<span class="default">$2 <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $3 <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; $3 <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default">$3 <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span>,<span class="p1">(<span class="default">$3 <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default">$2 <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">$2 <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default">$3 <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span>,<span class="p2">(<span class="default">$3 <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) );<br/>
<a NAME="E1:1_3"/><span class="lab"><font color="Green" title="E11">A27</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">u</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>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  ex <font color="Olive" title="b3">v</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>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">u</font>,<font color="Olive" title="b3">v</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" 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="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>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  ex <font color="Olive" title="b2">v</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>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">n</font>,<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c8">u</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>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</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>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c8">u</font>,<font color="Olive" title="b1">v</font>]</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c9">v1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c10">v2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E2:1_3_1"/><span class="lab"><font color="Green" title="E12">A28</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c8">u</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> ,<font color="Maroon" title="c8">u</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> ,<font color="Maroon" title="c9">v1</font>,<font color="Maroon" title="c10">v2</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1"><span class="lab"><font color="Green" title="E10">A11</font></span></a></span>;<br/>
<span class="kw">take </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">v1</font>,<font color="Maroon" title="c10">v2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c8">u</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">v1</font>,<font color="Maroon" title="c10">v2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]</span><br/>

<a NAME="E3:1_3_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">v1</font>,<font color="Maroon" title="c10">v2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:1_3_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c8">u</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">v1</font>,<font color="Maroon" title="c10">v2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_3_1"><span class="lab"><font color="Green" title="E12">A28</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>; <a class="txt" 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">consider </span><font color="Maroon" title="c7">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<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>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><span class="kw"> such that </span><br/><a NAME="E3:1_3"/><span class="lab"><font color="Green" title="E12">A29</font></span>: 
( <font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">z0</font>,<font color="Maroon" title="c6">t0</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="#E1:1_3"><span class="lab"><font color="Green" title="E11">A27</font></span></a>);<br/></span>
<span class="kw">take </span>
 <a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r0</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z0</font> &amp; <font color="Olive" title="b1">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t0</font> &amp; (  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b1">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) )</span><br/>

<span class="kw">take </span>
 <a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z0</font> &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t0</font> &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) )</span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D5" title="FUNCT_2:def.5">FUNCT_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c4">z0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_3"><span class="lab"><font color="Green" title="E12">A29</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t0</font> &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) )</span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D6" title="FUNCT_2:def.6">FUNCT_2:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c6">t0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_3"><span class="lab"><font color="Green" title="E12">A29</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c8">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c8">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]</span><br/><a NAME="E1:1_3_4"/><span class="lab"><font color="Green" title="E13">A30</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D5" title="FUNCT_2:def.5">FUNCT_2:def 5</a>, <a class="ref" href="funct_2.html#D6" title="FUNCT_2:def.6">FUNCT_2:def 6</a></span>;<br/><a NAME="E2:1_3_4"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c8">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D5" title="FUNCT_2:def.5">FUNCT_2:def 5</a>, <a class="ref" href="funct_2.html#D6" title="FUNCT_2:def.6">FUNCT_2:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_3_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c8">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_3"><span class="lab"><font color="Green" title="E12">A29</font></span></a>, <a class="txt" href="#E1:1_3_4"><span class="lab"><font color="Green" title="E13">A30</font></span></a></span>; <a class="txt" 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">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x0</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c8">r0</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E24:1"/><span class="lab"><font color="Green" title="E11">A31</font></span>: 
( <font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z0</font> &amp; <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t0</font> )
 <span class="kw">and </span><br/><a NAME="E25:1"/><span class="lab"><font color="Green" title="E12">A32</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>
<a NAME="E26:1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
;<br/>
<a NAME="E27:1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A33</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">t0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1"><span class="lab"><font color="Green" title="E6">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></span>;<br/>
<a NAME="E28:1"/><span class="lab"><font color="Green" title="E14">A34</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b1">n</font></span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 &amp; <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> $1</span>)</span> &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> $1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> $1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  );<br/>
<div><span class="lab"><font color="Green" title="E15">A35</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">n</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:1_4_1"/><span class="lab"><font color="Green" title="E16">A36</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><a NAME="E2:1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A37</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1"><span class="lab"><font color="Green" title="E12">A32</font></span></a></span>;<br/><a NAME="E3:1_4_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1"><span class="lab"><font color="Green" title="E12">A32</font></span></a>, <a class="txt" href="#E1:1_4_1"><span class="lab"><font color="Green" title="E16">A36</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_4_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1"><span class="lab"><font color="Green" title="E12">A32</font></span></a>, <a class="txt" href="#E2:1_4_1"><span class="lab"><font color="Green" title="E17">A37</font></span></a></span>; <a class="txt" 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>
<a NAME="E2:1_4"/><span class="lab"><font color="Green" title="E16">A38</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1"><span class="lab"><font color="Green" title="E13">A33</font></span></a>, <a class="txt" href="#E16:1"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="txt" href="#E20:1"><span class="lab"><font color="Green" title="E9">A10</font></span></a>, <a class="txt" href="#E24:1"><span class="lab"><font color="Green" title="E11">A31</font></span></a>, <a class="txt" href="#E25:1"><span class="lab"><font color="Green" title="E12">A32</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:1_4"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:1_4"><span class="lab"><font color="Green" title="E16">A38</font></span></a>, <a class="txt" href="#E1:1_4"><span class="lab"><font color="Green" title="E15">A35</font></span></a>);</span> <a class="txt" 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>
<a NAME="E29:1"/><span class="lab"><font color="Green" title="E15">A39</font></span>: 
 for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span></span>)</span></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> $1</span>)</span></span>)</span></span>)</span>;<br/>
<div><span class="lab"><font color="Green" title="E16">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:1_5_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><a NAME="E2:1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A41</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>)</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> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:1_5_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="#E28:1"><span class="lab"><font color="Green" title="E14">A34</font></span></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/><a NAME="E4:1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A42</font></span>: 
<font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><a NAME="E5:1_5_1"/>
<font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1"><span class="lab"><font color="Green" title="E14">A34</font></span></a></span>;<br/><a NAME="E6:1_5_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/><a NAME="E7:1_5_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></span>;<br/><a NAME="E8:1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A43</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>;<br/><a NAME="E9:1_5_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1"><span class="lab"><font color="Green" title="E14">A34</font></span></a></span>;<br/><a NAME="E10:1_5_1"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_5_1"><span class="lab"><font color="Green" title="E18">A42</font></span></a>, <a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><a NAME="E11:1_5_1"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_5_1"><span class="lab"><font color="Green" title="E19">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A44</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E13:1_5_1"/>
2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T8" title="NEWTON:th.8">NEWTON:8</a></span>;<br/><span class="kw">then </span>1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T79" title="XCMPLX_1:th.79">XCMPLX_1:79</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A45</font></span>: <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></span>
;<br/><a NAME="E16:1_5_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>)</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> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></span>;<br/><a NAME="E17:1_5_1"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_5_1"><span class="lab"><font color="Green" title="E17">A41</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E18:1_5_1"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_5_1"><span class="lab"><font color="Green" title="E20">A44</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E19:1_5_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:1_5_1"><span class="lab"><font color="Green" title="E21">A45</font></span></a>, <a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>; <a class="txt" 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>
<a NAME="E2:1_5"/>
2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>;<br/>
<a NAME="E3:1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A46</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <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/>
<a NAME="E4:1_5"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E3:1_5"><span class="lab"><font color="Green" title="E17">A46</font></span></a>, <a class="txt" href="#E1:1_5"><span class="lab"><font color="Green" title="E16">A40</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:1_5"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" 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="lab"><font color="Green" title="E16">A47</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span><br/><a NAME="E1:1_6_1"/><span class="lab"><font color="Green" title="E17">A48</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:1"><span class="lab"><font color="Green" title="E15">A39</font></span></a></span>;<br/><a NAME="E2:1_6_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T83" title="NEWTON:th.83">NEWTON:83</a></span>;<br/><a NAME="E3:1_6_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T139" title="XREAL_1:th.139">XREAL_1:139</a></span>;<br/><a NAME="E4:1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A49</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E5:1_6_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T83" title="NEWTON:th.83">NEWTON:83</a></span>;<br/><a NAME="E6:1_6_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T139" title="XREAL_1:th.139">XREAL_1:139</a></span>;<br/><a NAME="E7:1_6_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_6_1"><span class="lab"><font color="Green" title="E18">A49</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E8:1_6_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_6_1"><span class="lab"><font color="Green" title="E17">A48</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" 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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font> implies  ex <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font> )</span><br/><span class="kw">assume </span><a NAME="E1:1_7"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/> for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">p1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:1_7"/><span class="lab"><font color="Green" title="E17">A50</font></span>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">p1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T92" title="PREPOWER:th.92">PREPOWER:92</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c11">p</font> = <font color="Maroon" title="c10">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">p</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font></span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c13">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">p</font> implies  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_7_1"/><span class="lab"><font color="Green" title="E18">A51</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">and </span><br/><a NAME="E2:1_7_1"/><span class="lab"><font color="Green" title="E19">A52</font></span>: 
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c14">k1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:1_7_1"/><span class="lab"><font color="Green" title="E20">A53</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_7_1"><span class="lab"><font color="Green" title="E18">A51</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">k1</font> = <font color="Maroon" title="c14">k1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E6:1_7_1"/>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_7_1"><span class="lab"><font color="Green" title="E20">A53</font></span></a></span>;<br/><a NAME="E7:1_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A54</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c11">p</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c11">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:1"><span class="lab"><font color="Green" title="E16">A47</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">k2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:1_7_1"/><span class="lab"><font color="Green" title="E22">A55</font></span>: 
<font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_7_1"><span class="lab"><font color="Green" title="E19">A52</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">k2</font> = <font color="Maroon" title="c16">k2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="lab"><font color="Green" title="E23">A56</font></span>: 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c11">p</font></span>)</span> = 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">p1</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></span>
;<br/><a NAME="E12:1_7_1"/>
<font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_7_1"><span class="lab"><font color="Green" title="E22">A55</font></span></a></span>;<br/><a NAME="E13:1_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A57</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c11">p</font></span>)</span>)</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> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c11">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c11">p</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:1"><span class="lab"><font color="Green" title="E16">A47</font></span></a>, <a class="txt" href="#E7:1_7_1"><span class="lab"><font color="Green" title="E21">A54</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E14:1_7_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c11">p</font></span>)</span>)</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> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c11">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></span>;<br/><a NAME="E15:1_7_1"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c10">p1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_7_1"><span class="lab"><font color="Green" title="E24">A57</font></span></a>, <a class="txt" href="#E11:1_7_1"><span class="lab"><font color="Green" title="E23">A56</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E16:1_7_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_7"><span class="lab"><font color="Green" title="E17">A50</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" 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>
<a NAME="E32:1"/><span class="kw">then </span>
<font color="Maroon" title="c7">x0</font> is  <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tbsp_1.html#D4" title="TBSP_1:def.4">TBSP_1:def 4</a></span>;<br/>
<a NAME="E33:1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A58</font></span>: 
<font color="Maroon" title="c7">x0</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="ref" href="tbsp_1.html#D5" title="TBSP_1:def.5">TBSP_1:def 5</a></span>;<br/>
<a NAME="E34:1"/><span class="lab"><font color="Green" title="E18">A59</font></span>: 
 for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>)</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>);<br/>
<div><span class="lab"><font color="Green" title="E19">A60</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_8_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:1_8_1"/><span class="lab"><font color="Green" title="E20">A61</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><a NAME="E2:1_8_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1"><span class="lab"><font color="Green" title="E14">A34</font></span></a></span>;<br/><a NAME="E3:1_8_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T216" title="XREAL_1:th.216">XREAL_1:216</a></span>;<br/><a NAME="E4:1_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A62</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#T1" title="PCOMPS_1:th.1">PCOMPS_1:1</a></span>;<br/><a NAME="E5:1_8_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 ;<br/><a NAME="E6:1_8_1"/><span class="kw">then </span>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1"><span class="lab"><font color="Green" title="E14">A34</font></span></a></span>;<br/><a NAME="E7:1_8_1"/><span class="kw">then </span>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_8_1"><span class="lab"><font color="Green" title="E21">A62</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:1_8_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_8_1"><span class="lab"><font color="Green" title="E20">A61</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" 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>
<a NAME="E2:1_8"/><span class="lab"><font color="Green" title="E20">A63</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1"><span class="lab"><font color="Green" title="E14">A34</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:1_8"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:1_8"><span class="lab"><font color="Green" title="E20">A63</font></span></a>, <a class="txt" href="#E1:1_8"><span class="lab"><font color="Green" title="E19">A60</font></span></a>);</span> <a class="txt" 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="lab"><font color="Green" title="E19">A64</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span>)</span><br/><span class="kw">set </span><font color="Maroon" title="c10">m1</font> = <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><a NAME="E1:1_9"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1"><span class="lab"><font color="Green" title="E14">A34</font></span></a></span>;<br/><a NAME="E2:1_9"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T215" title="XREAL_1:th.215">XREAL_1:215</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:1_9"/><span class="lab"><font color="Green" title="E20">A65</font></span>: 
 for <font color="Olive" title="b1">l</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c11">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">l</font></span>)</span>,<span class="p1">(<span class="default"><a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1"><span class="lab"><font color="Green" title="E17">A58</font></span></a>, <a class="ref" href="tbsp_1.html#D3" title="TBSP_1:def.3">TBSP_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">n</font> =  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> (<font color="Maroon" title="c11">n1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/><a NAME="E6:1_9"/><span class="lab"><font color="Green" title="E21">A66</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_9"><span class="lab"><font color="Green" title="E20">A65</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:1_9"/><span class="lab"><font color="Green" title="E22">A67</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/><a NAME="E9:1_9"/>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1"><span class="lab"><font color="Green" title="E14">A34</font></span></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/><a NAME="E10:1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A68</font></span>: 
<font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">k</font> = <font color="Maroon" title="c13">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E12:1_9"/>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_9"><span class="lab"><font color="Green" title="E22">A67</font></span></a></span>;<br/><a NAME="E13:1_9"/><span class="kw">then </span>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1"><span class="lab"><font color="Green" title="E18">A59</font></span></a></span>;<br/><a NAME="E14:1_9"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_9"><span class="lab"><font color="Green" title="E23">A68</font></span></a>, <a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><a NAME="E15:1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A69</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p2">(<span class="default"><a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>)</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> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_9"><span class="lab"><font color="Green" title="E21">A66</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E16:1_9"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p2">(<span class="default"><a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>)</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> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></span>;<br/><a NAME="E17:1_9"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<span class="p1">(<span class="default"><a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:1_9"><span class="lab"><font color="Green" title="E24">A69</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E18:1_9"/>
 <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">m</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>; <a class="txt" 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="lab"><font color="Green" title="E20">A70</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_10"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span><br/><span class="kw">thus </span><a NAME="E1:1_10"/>
 not  <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:1_10_1"/><span class="lab"><font color="Green" title="E21">A71</font></span>: 
 <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:1_10_1"/>
 <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:1"><span class="lab"><font color="Green" title="E19">A64</font></span></a></span>;<br/>
<a NAME="E3:1_10_1"/><span class="kw">then </span>
 <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">r0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_10_1"><span class="lab"><font color="Green" title="E21">A71</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:1_10_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1"><span class="lab"><font color="Green" title="E14">A34</font></span></a></span>; <a class="txt" 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>
<a NAME="E37:1"/>
 not  <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:1_11"/>
 <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:1_11"/><span class="lab"><font color="Green" title="E21">A72</font></span>: 
 <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">and </span><br/><a NAME="E4:1_11"/><span class="lab"><font color="Green" title="E22">A73</font></span>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:1_11"/>
 ex <font color="Olive" title="b1">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_11"><span class="lab"><font color="Green" title="E22">A73</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:1_11"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:1"><span class="lab"><font color="Green" title="E20">A70</font></span></a>, <a class="txt" href="#E3:1_11"><span class="lab"><font color="Green" title="E21">A72</font></span></a></span>; <a class="txt" 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="E38:1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E2">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
