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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">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<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">D</font>,<a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">R</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>







<b>assume </b><a NAME="E1:49"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">d</font> <a href="rfunct_3.html#R1" target="_self">is_common_for_dom</a> <font color="Maroon">f</font>
 ;<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">D</font>,<a href="numbers.html#K1">REAL</a> ; <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">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="E2:49"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">n</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">R</font> <a href="rfunct_3.html#K17" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_3.html#K17" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> )
 ;<br/>

<b>set </b><font color="Maroon">m</font> =  <a href="rfunct_3.html#K1" target="_self">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>;<br/>
<a NAME="E3:49"/><i><font color="Green">E32</font></i>: 
(  <a href="rfunct_3.html#K1" target="_self">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> &amp;  <a href="rfunct_3.html#K1" target="_self">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="square_1.html#T35">SQUARE_1:35</a></i>;<br/>
<a NAME="E4:49"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="rfunct_3.html#K17" target="_self">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K1" target="_self">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:49"/><b>then </b><i><font color="Green">E33</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rfunct_3.html#K1" target="_self">min</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:49"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T2" target="_self">Th2</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E7:49"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">R</font> &amp; <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="ref" href="rfunct_3.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as    <a href="rfunct_3.html#M2" target="_self">Element</a> of  <a href="rfunct_3.html#K4" target="_self">PFuncs</a> <font color="Maroon">D</font>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="rfunct_3.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E10:49"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="rfunct_3.html#K13" target="_self">(#)</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:49"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E10:49"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rfunct_3.html#T4" target="_self">Th4</a>, <a class="ref" href="rfunct_3.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E12:49"/>
<font color="Maroon">d</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="ref" href="rfunct_3.html#T5" target="_self">Th5</a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>


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