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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">pmet</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; ( for <font color="Olive" title="b2">pmet'</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>  st <font color="Olive" title="b1">pmet</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">pmet'</font> holds <br/><font color="Olive" title="b2">pmet'</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> ) holds <br/>for <font color="Olive" title="b2">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b3">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b2">A</font> holds <br/><span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Olive" title="b1">pmet</font>,<font color="Olive" title="b2">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><b>let </b><font color="Maroon" title="c2">pmet</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; ( for <font color="Olive" title="b1">pmet'</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>  st <font color="Maroon" title="c2">pmet</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">pmet'</font> holds <br/><font color="Olive" title="b1">pmet'</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> ) implies for <font color="Olive" title="b1">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font> holds <br/><span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Olive" title="b1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:20"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c2">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E2:20"/><i><font color="Green" title="E22">A2</font></i>: 
for <font color="Olive" title="b1">pmet'</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c1">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>  st <font color="Maroon" title="c2">pmet</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">pmet'</font> holds <br/><font color="Olive" title="b1">pmet'</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font> holds <br/><span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Olive" title="b1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<b>let </b><font color="Maroon" title="c3">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A</font> holds <br/><span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><b>let </b><font color="Maroon" title="c4">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A</font> implies <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>



<b>set </b><font color="Maroon" title="c5">rn</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Z</font> = <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>, <font color="Maroon" title="c7">infA</font> = <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">InfA</font> =  <a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c3">A</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>assume </b><a NAME="E5:20"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E6:20"/>
<font color="Maroon" title="c7">infA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">infa</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">infa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">infA</font> or <font color="Maroon" title="c9">infa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font> )</span><br/>

<b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon" title="c9">infa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">infA</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">infa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font></span><br/>

<b>then consider </b><font color="Maroon" title="c10">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:20_1"/><i><font color="Green" title="E24">A4</font></i>: 
( <font color="Maroon" title="c10">a</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="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c3">A</font></span>)</span> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c9">infa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:20_1"/>
<font color="Maroon" title="c9">infa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="#E3:20_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="nagata_2.html#T6" target="_self" title="NAGATA_2:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E5:20_1"/>
<font color="Maroon" title="c9">infa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:20"/><b>then </b><i><font color="Green" title="E24">A5</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">infA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">Z</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E8:20"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds   <a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Olive" title="b1">p</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 
<b>by </b><i><a class="txt" href="#E2:20"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="nagata_2.html#T4" target="_self" title="NAGATA_2:th.4">Th4</a></i>;<br/>
<a NAME="E9:20"/><b>then </b>
(  <a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c3">A</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  )
 
<b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="nagata_2.html#T8" target="_self" title="NAGATA_2:th.8">Th8</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E10:20"/><b>then </b>
( <font color="Maroon" title="c8">InfA</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c6">Z</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T10" title="PCOMPS_1:th.10">PCOMPS_1:10</a>, <a class="ref" href="topreal6.html#T83" title="TOPREAL6:th.83">TOPREAL6:83</a></i>;<br/>
<a NAME="E11:20"/><b>then </b>
( <font color="Maroon" title="c8">InfA</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">InfA</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">A</font></span>)</span> &amp; <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">Z</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c3">A</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">T</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a>, <a class="ref" href="tops_2.html#T57" title="TOPS_2:th.57">TOPS_2:57</a></i>;<br/>
<a NAME="E12:20"/><b>then </b>
( <font color="Maroon" title="c8">InfA</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Z</font> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E5:20"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="#E7:20"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E13:20"/>
<span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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