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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="finseq_1.html#NM1">FinSequence</a>] means ( <font color="Maroon">a<sub>1</sub></font> is <a href="funcop_1.html#V1">Function-yielding</a> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E1:34_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i><a class="txt" href="funct_7.html#E1:34"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E2:34_1"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:34_1_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="funcop_1.html#V1">Function-yielding</a> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>] )
 <b>and </b><br/><a NAME="E2:34_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> is <a href="funcop_1.html#V1">Function-yielding</a>
 ;<br/>

<a NAME="E3:34_1_1"/><i><font color="Green">E16</font></i>: 
( <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E4:34_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="funcop_1.html#V1">Function-yielding</a> &amp; <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> is <a href="funcop_1.html#V1">Function-yielding</a> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="funct_7.html#E2:34_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_7.html#T25" target="_self">Th25</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:34_1_1"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>] &amp; <font color="Maroon">c<sub>2</sub></font> is   <a href="funct_1.html#NM1">Function</a> )
 
<b>by </b><i><a class="txt" href="funct_7.html#E1:34_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="funct_7.html#E3:34_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:34_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="funct_7.html#E2:34"><i><font color="Green">E11</font></i></a>, <a class="txt" href="funct_7.html#E4:34_1_1"><i><font color="Green">E17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:34_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="finseq_1.html#S3">FINSEQ_1:sch 3</a>(<a class="txt" href="funct_7.html#E1:34_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="funct_7.html#E2:34_1"><i><font color="Green">E13</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:34_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> holds <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


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