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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">p</font> being   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> );<br/>
<span class="kw">consider </span><font color="Maroon" title="c1">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E2:78_1"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" onmouseover="rs('xboole_0/S1')" onmouseout="rh()">XBOOLE_0:sch 1</a>();<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c1">Y</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> iff  ex <font color="Olive" title="b2">p</font> being   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> st <br/>( <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>] &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> ) )</span><br/>

<a NAME="E3:78_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> holds <br/> ex <font color="Olive" title="b2">p</font> being   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> st <br/>( <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>] &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> implies  ex <font color="Olive" title="b1">p</font> being   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:78_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> )</span><br/>

<a NAME="E2:78_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  &amp;  ex <font color="Olive" title="b1">p</font> being   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78_1"><span class="lab"><font color="Green" title="E31">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:78_1_1"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:78_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> iff  ex <font color="Olive" title="b2">p</font> being   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> st <br/>( <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>] &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78_1"><span class="lab"><font color="Green" title="E31">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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