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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="E1:5"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c2">S</font> is <a href="kurato_2.html#V1" title="KURATO_2:attr.1">descending</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/>
<div><i><font color="Green" title="E6">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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> ;<br/><a NAME="E1:5_1"/>
( <font color="Maroon" title="c2">S</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">S</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">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="c4">n</font> &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="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <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>, <a class="ref" href="kurato_2.html#T22" title="KURATO_2:th.22">KURATO_2:22</a></i>;<br/><a NAME="E2:5_1"/><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="c4">n</font> <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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"><span class="p2">(<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 <b>by </b><i><a class="ref" href="tbsp_1.html#T32" title="TBSP_1:th.32">TBSP_1:32</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E3:5_1"/>
<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="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">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> ;<br/><b>assume </b><a NAME="E1:5_2"/><i><font color="Green" title="E7">A3</font></i>: 
( <font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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> &amp; <font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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> &amp; <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> )
 ;<br/><a NAME="E2:5_2"/>
( <font color="Maroon" title="c2">S</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c5">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c2">S</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">m</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="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c5">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="c5">n</font> &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="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">m</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="c4">m</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="compl_sp.html#E1:5_2"><i><font color="Green" title="E7">A3</font></i></a>, <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>, <a class="ref" href="kurato_2.html#T22" title="KURATO_2:th.22">KURATO_2:22</a></i>;<br/><b>hence </b><a NAME="E3:5_2"/>
<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="c5">n</font> <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="c4">m</font>
 <b>by </b><i><a class="ref" href="tbsp_1.html#T32" title="TBSP_1:th.32">TBSP_1:32</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:5"/>
(  <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#V1" title="SEQ_2:attr.1">bounded_above</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#E2:5"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="seqm_3.html#D4" title="SEQM_3:def.4">SEQM_3:def 4</a>, <a class="ref" href="seq_2.html#D3" title="SEQ_2:def.3">SEQ_2:def 3</a></i>;<br/>


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