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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="seqfunc.html#M1" target="_self">Functional_Sequence</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:54"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:54"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="seqfunc.html#D10" target="_self">Def10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:54_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seqfunc.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E2:54_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seqfunc.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:54_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:54_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E4:54_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E5:54_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K6" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="seqfunc.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:54"/>
<font color="Maroon">c<sub>4</sub></font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K6" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:54"><i><font color="Green">E39</font></i></a>, <a class="ref" href="seqfunc.html#D10" target="_self">Def10</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:54_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seqfunc.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E2:54_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seqfunc.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:54_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:54_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E4:54_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:54_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="seqfunc.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:54"/>
<font color="Maroon">c<sub>4</sub></font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:54"><i><font color="Green">E39</font></i></a>, <a class="ref" href="seqfunc.html#D10" target="_self">Def10</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:54_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seqfunc.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E2:54_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seqfunc.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:54_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:54_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E4:54_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:54_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K8" target="_self">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="seqfunc.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:54"/>
<font color="Maroon">c<sub>4</sub></font> <a href="seqfunc.html#R1" target="_self">common_on_dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="seqfunc.html#K8" target="_self">(#)</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="seqfunc.html#E2:54"><i><font color="Green">E39</font></i></a>, <a class="ref" href="seqfunc.html#D10" target="_self">Def10</a></i>;<br/>


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