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<a NAME="E1:72_1_1"/>
 <a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c3">S</font> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">Si</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <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="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font></span><br/><b>reconsider </b><font color="Maroon" title="c5">DSeq</font> = <font color="Maroon" title="c3">S</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c4">n</font> as   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font> ;<br/><b>reconsider </b><font color="Maroon" title="c6">n1</font> = <font color="Maroon" title="c4">n</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:72_1_1_1_1"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c5">DSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">DSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font></span><br/>

<a NAME="E1:72_1_1_1_1_1"/><i><font color="Green" title="E54">Y</font></i>: 
<font color="Maroon" title="c5">DSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">n1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></i>;<br/>
<a NAME="E2:72_1_1_1_1_1"/><i><font color="Green" title="E55">X</font></i>: 
<font color="Maroon" title="c3">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">n1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T52" title="NAT_1:th.52">NAT_1:52</a></i>;<br/>
<a NAME="E3:72_1_1_1_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:72_1_1_1_1_1"/>
<font color="Maroon" title="c5">DSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 <b>by </b><i><a class="txt" href="#E2:72_1_1_1_1_1"><i><font color="Green" title="E55">X</font></i></a>, <a class="txt" href="#E1:72_1_1_1_1_1"><i><font color="Green" title="E54">Y</font></i></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="E4:72_1_1_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">DSeq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></i>;<br/><a NAME="E5:72_1_1_1_1"/><b>then </b><i><font color="Green" title="E54">B4</font></i>: 
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c5">DSeq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D8" title="PROB_1:def.8">PROB_1:def 8</a></i>;<br/><a NAME="E6:72_1_1_1_1"/>
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c5">DSeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">DSeq</font></span>)</span>
 <b>by </b><i><a class="ref" href="setlim_1.html#T8" target="_self" title="SETLIM_1:th.8">Th16</a></i>;<br/><a NAME="E7:72_1_1_1_1"/><b>then </b>
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c5">DSeq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="setlim_1.html#T6" target="_self" title="SETLIM_1:th.6">Th11</a></i>;<br/><b>hence </b><a NAME="E8:72_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 <b>by </b><i><a class="txt" href="#E5:72_1_1_1_1"><i><font color="Green" title="E54">B4</font></i></a>, <a class="ref" href="setlim_1.html#D2" target="_self" title="SETLIM_1:def.2">def2</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="E2:72_1_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Si</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></i>;<br/>
<b>hence </b><a NAME="E3:72_1_1_1"/>
 <a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c3">S</font> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">Si</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</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>
<b>hence </b><a NAME="E2:72_1_1"/>
 <a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c3">S</font> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">Si</font>
 ; <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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