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<span class="kw">let </span><font color="Maroon" title="c1">u</font> be   <a href="valued_0.html#V5" title="VALUED_0:attr.5">integer-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being   <a href="int_6.html#NM1" title="INT_6:NM.1">CR_Sequence</a>  st  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"> { <span class="default"> <font color="Olive" title="b2">z</font> where <font color="Olive" title="b2">z</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b2">z</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c1">u</font>,<font color="Olive" title="b1">m</font> </span> } </span>  is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">m</font> be   <a href="int_6.html#NM1" title="INT_6:NM.1">CR_Sequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">m</font> implies <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">z</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c2">m</font> </span> } </span>  is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">X</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">x</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c2">m</font> </span> } </span> ;<br/>
<a NAME="E1:34"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">x</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c2">m</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">x</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c2">m</font> </span> } </span> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">x</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c2">m</font> </span> } </span>  or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">x</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c2">m</font> </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:34_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">x</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c2">m</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">x</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c2">m</font> </span> } </span> </span><br/>

<a NAME="E2:34_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c2">m</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:34_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">x</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c2">m</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:34"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">m</font> implies <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">z</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c2">m</font> </span> } </span>  is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T31" target="_self" onmouseover="rs('number14/T31')" onmouseout="rh()">Th31</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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