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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">H</font> be    <a href="seqfunc.html#M1" target="_self">Functional_Sequence</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/>





<b>assume </b><a NAME="E1:46"/><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="seqfunc.html#R1" target="_self">common_on_dom</a> <font color="Maroon">H</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:46_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E2:46_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:46_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:46_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K5" target="_self">abs</a> <font color="Maroon">H</font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K5" target="_self">abs</a> <font color="Maroon">H</font></span>)</span> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_1.html#K21">abs</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:46"/>
<span class="p1">(<span class="default"><a href="seqfunc.html#K5" target="_self">abs</a> <font color="Maroon">H</font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:46_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E2:46_2"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:46_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:46_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K4" target="_self">-</a> <font color="Maroon">H</font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K4" target="_self">-</a> <font color="Maroon">H</font></span>)</span> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T14">SEQ_1:14</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:46"/>
<span class="p1">(<span class="default"><a href="seqfunc.html#K4" target="_self">-</a> <font color="Maroon">H</font></span>)</span> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="seq_1.html#K17">-</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K10" target="_self">#</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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