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<b>let </b><font color="Maroon">D1</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">D2</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">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="rfunct_3.html#K4" target="_self">PFuncs</a> <font color="Maroon">D1</font>,<font color="Maroon">D2</font>;<br/><b>let </b><font color="Maroon">d1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:44"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">d1</font> <a href="rfunct_3.html#R1" target="_self">is_common_for_dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<b>let </b><font color="Maroon">G</font> be    <a href="rfunct_3.html#M2" target="_self">Element</a> of  <a href="rfunct_3.html#K4" target="_self">PFuncs</a> <font color="Maroon">D1</font>,<font color="Maroon">D2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_3.html#D9" target="_self">RFUNCT_3:def 9</a><br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E2:44"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1"><b>case </b></a><a NAME="E1:44_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:44_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:44_1_1_1"/>
<font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:44"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="rfunct_3.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_2"><b>case </b></a><a NAME="E1:44_1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:44_1_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
(  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:44_1_1_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:44"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E4:44_1_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:44_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:44_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:44_1_1_2"/><b>then </b>
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:44_1_1_2"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/><b>hence </b><a NAME="E7:44_1_1_2"/>
<font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:44"><i><font color="Green">E31</font></i></a>, <a class="ref" href="rfunct_3.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:44"/>
<font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
 ;<br/>


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