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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  <a href="compl_sp.html#V1" title="COMPL_SP:attr.1">bounded</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">M</font>;<br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c2">S</font> is <a href="kurato_2.html#V1" title="KURATO_2:attr.1">descending</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>

<b>set </b><font color="Maroon" title="c3">d</font> =  <a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c2">S</font>;<br/>
<b>let </b><font color="Maroon" title="c4">F</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">M</font>;<br/>

<b>assume </b><a NAME="E2:7"/><i><font color="Green" title="E7">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c4">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Olive" title="b1">i</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="tbsp_1.html#D5" title="TBSP_1:def.5">TBSP_1:def 5</a><br/>

<b>assume </b><a NAME="E3:7"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E4:7"/>
(  <a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c2">S</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> &amp;  <a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c2">S</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> )
 
<b>by </b><i><a class="txt" href="compl_sp.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="compl_sp.html#T1" target="_self" title="COMPL_SP:th.1">Th1</a>, <a class="ref" href="compl_sp.html#T2" target="_self" title="COMPL_SP:th.2">Th2</a></i>;<br/>
<a NAME="E5:7"/><b>then </b>
 <a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c2">S</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T52" title="SEQ_4:th.52">SEQ_4:52</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:7"/><i><font color="Green" title="E9">A4</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="compl_sp.html#E3:7"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">n</font>
;<br/>

<b>let </b><font color="Maroon" title="c7">m1</font>, <font color="Maroon" title="c8">m2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E8:7"/><i><font color="Green" title="E10">A5</font></i>: 
( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m1</font> &amp; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m2</font> )
 ;<br/>

<a NAME="E9:7"/>
( <font color="Maroon" title="c2">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c7">m1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c2">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c8">m2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c7">m1</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c8">m2</font> )
 
<b>by </b><i><a class="txt" href="compl_sp.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="compl_sp.html#E2:7"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="compl_sp.html#E8:7"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="kurato_2.html#T22" title="KURATO_2:th.22">KURATO_2:22</a></i>;<br/>
<a NAME="E10:7"/><b>then </b>
( <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c2">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c6">n</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> &amp;  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font> )
 
<b>by </b><i><a class="ref" href="compl_sp.html#D1" target="_self" title="COMPL_SP:def.1">def1</a>, <a class="ref" href="compl_sp.html#D2" target="_self" title="COMPL_SP:def.2">def2</a></i>;<br/>
<a NAME="E11:7"/><b>then </b><i><font color="Green" title="E11">A5</font></i>: 
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> )
 
<b>by </b><i><a class="txt" href="compl_sp.html#E7:7"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="tbsp_1.html#D10" title="TBSP_1:def.10">TBSP_1:def 10</a>, <a class="ref" href="tbsp_1.html#T29" title="TBSP_1:th.29">TBSP_1:29</a></i>;<br/>
<a NAME="E12:7"/><b>then </b>
<span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E13:7"/>
not <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c8">m2</font></span>)</span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E11:7"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div>
