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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font> holds <br/> (  <a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlcon</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="finseq_8.html#K3" title="FINSEQ_8:func.3">ovlpart</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="finseq_8.html#K5" title="FINSEQ_8:func.5">ovlldiff</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="finseq_8.html#K6" title="FINSEQ_8:func.6">ovlrdiff</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlcon</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="finseq_8.html#K3" title="FINSEQ_8:func.3">ovlpart</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="finseq_8.html#K5" title="FINSEQ_8:func.5">ovlldiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="finseq_8.html#K6" title="FINSEQ_8:func.6">ovlrdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>



<a NAME="E1:19"/><i><font color="Green" title="E10">A1</font></i>: 
(  <a href="finseq_8.html#K3" title="FINSEQ_8:func.3">ovlpart</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c2">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="finseq_8.html#K3" title="FINSEQ_8:func.3">ovlpart</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font></span>)</span></span>)</span> &amp;  <a href="finseq_8.html#K3" title="FINSEQ_8:func.3">ovlpart</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="finseq_8.html#K3" title="FINSEQ_8:func.3">ovlpart</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_8.html#D2" target="_self" title="FINSEQ_8:def.2">Def2</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1

;<br/>
<a NAME="E3:19"/><b>then </b>
 <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c2">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
;<br/>
<a NAME="E4:19"/><b>then </b><i><font color="Green" title="E11">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K3" title="FINSEQ_8:func.3">ovlpart</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_8.html#D2" target="_self" title="FINSEQ_8:def.2">Def2</a></i>;<br/>
<b>then </b><i><font color="Green" title="E12">A3</font></i>:  <a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlcon</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> = 
<font color="Maroon" title="c2">f</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_8.html#T2" target="_self" title="FINSEQ_8:th.2">Th2</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
;<br/>
<i><font color="Green" title="E13">A4</font></i>:  <a href="finseq_8.html#K5" title="FINSEQ_8:func.5">ovlldiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> = 
<font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="finseq_8.html#T6" target="_self" title="FINSEQ_8:th.6">Th6</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 

;<br/>
 <a href="finseq_8.html#K6" title="FINSEQ_8:func.6">ovlrdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> = 
 <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font>
<b>by </b><i><a class="txt" href="#E4:19"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="finseq_8.html#T2" target="_self" title="FINSEQ_8:th.2">Th2</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 

;<br/>
<b>hence </b><a NAME="E8:19"/>
(  <a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlcon</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="finseq_8.html#K3" title="FINSEQ_8:func.3">ovlpart</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="finseq_8.html#K5" title="FINSEQ_8:func.5">ovlldiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="finseq_8.html#K6" title="FINSEQ_8:func.6">ovlrdiff</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E4:19"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="#E5:19"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="#E6:19"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="finseq_8.html#T6" target="_self" title="FINSEQ_8:th.6">Th6</a></i>; <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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