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<b>let </b><font color="Maroon" title="c1">q</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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">p</font> being   <a href="funct_7.html#M2" title="FUNCT_7:mode.2">FuncSequence</a> of <font color="Maroon" title="c1">q</font> holds <br/> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Olive" title="b1">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Olive" title="b1">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">p</font> be    <a href="funct_7.html#M2" title="FUNCT_7:mode.2">FuncSequence</a> of <font color="Maroon" title="c1">q</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font></span>)</span> )</span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1"><b>suppose </b></a><a NAME="E1:95_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font></span>)</span> )</span><br/><div class="add"><a NAME="E2:95_1_1"/><b>then </b>
(  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="funct_7.html#D10" target="_self" title="FUNCT_7:def.10">Def10</a>, <a class="ref" href="funct_7.html#T41" target="_self" title="FUNCT_7:th.41">Th41</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/><b>hence </b><a NAME="E3:95_1_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_2"><b>suppose </b></a><a NAME="E1:95_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font></span>)</span> )</span><br/><div class="add"><a NAME="E2:95_1_2"/><b>then </b>
(  <a href="funct_7.html#K6" title="FUNCT_7:func.6">firstdom</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_7.html#K7" title="FUNCT_7:func.7">lastrng</a> <font color="Maroon" title="c2">p</font> &amp;  <a href="funct_7.html#K7" title="FUNCT_7:func.7">lastrng</a> <font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_7.html#T61" target="_self" title="FUNCT_7:th.61">Th61</a>, <a class="ref" href="funct_7.html#T68" target="_self" title="FUNCT_7:th.68">Th68</a></i>;<br/><b>hence </b><a NAME="E3:95_1_2"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_7.html#T64" target="_self" title="FUNCT_7:th.64">Th64</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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