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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>] <span class="kw">means </span>( $1 is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  implies <font color="Maroon">P<sub>1</sub></font>[$1] );<br/>
<a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
for <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>]</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>] )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36_1_1"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
( <font color="Maroon" title="c1">p</font> is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>] )
 <span class="kw">and </span><br/><a NAME="E2:36_1_1"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>]</span><br/>

<a NAME="E3:36_1_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/>
<a NAME="E4:36_1_1"/>
( <font color="Maroon" title="c1">p</font> is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="funct_7.html#T25" target="_self" title="FUNCT_7:th.25">Th25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:36_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E1:36_1_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E3:36_1_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:36_1"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E11">A1</font></span></a></span>;<br/>
<a NAME="E3:36_1"/>
for <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S3" title="FINSEQ_1:sch.3">FINSEQ_1:sch 3</a>(<a class="txt" href="#E2:36_1"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:36_1"/>
for <font color="Olive" title="b1">p</font> being  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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