<?xml version="1.0"?>
<div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:69_1_1_1"/>
( <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  or  not <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_1_1"/>
<font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; (  for <font color="Olive" title="b2">t</font> being   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Olive" title="b2">t</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">Y</font> = <font color="Maroon" title="c2">Y</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c4">y</font> =  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c3">Y</font>;<br/><a NAME="E3:69_1_1_1_1"/>
 <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="qc_lang1.html#K1" title="QC_LANG1:func.1">QC-symbols</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T3" target="_self" onmouseover="rs('qc_lang1/T3')" onmouseout="rh()">Th3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">yp</font> =  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c3">Y</font> as   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">yp</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"> ( <font color="Maroon" title="c5">yp</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; (  for <font color="Olive" title="b1">t</font> being   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Maroon" title="c5">yp</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> ) )</span><br/><a NAME="E5:69_1_1_1_1"/>
 for <font color="Olive" title="b1">t</font> being   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">Y</font> holds <br/><font color="Maroon" title="c5">yp</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">t</font> be   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</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"> ( <font color="Maroon" title="c6">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">Y</font> implies <font color="Maroon" title="c5">yp</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_1_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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"> <font color="Maroon" title="c5">yp</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">t</font> = <font color="Maroon" title="c6">t</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a></span>;<br/>
<a NAME="E3:69_1_1_1_1_1"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c3">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="nat_1.html#D1" onmouseover="rs('nat_1/D1')" onmouseout="rh()">NAT_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:69_1_1_1_1_1"/>
<font color="Maroon" title="c5">yp</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="qc_lang1.html#D33" target="_self" onmouseover="rs('qc_lang1/D33')" onmouseout="rh()">Def33</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/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:69_1_1_1_1"/>
( <font color="Maroon" title="c5">yp</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; (  for <font color="Olive" title="b1">t</font> being   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Maroon" title="c5">yp</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D1" onmouseover="rs('nat_1/D1')" onmouseout="rh()">NAT_1:def 1</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_1_2"/>
 not <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; (  for <font color="Olive" title="b2">t</font> being   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Olive" title="b2">t</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:69_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp;  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )
 ;<br/><span class="kw">set </span><font color="Maroon" title="c4">R</font> =  the    <a href="qc_lang1.html#M2" title="QC_LANG1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font>;<br/><a NAME="E4:69_1_1_1_2"/>
(  the    <a href="qc_lang1.html#M2" title="QC_LANG1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K1" title="QC_LANG1:func.1">QC-symbols</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> &amp; <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_1_2"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="qc_lang1.html#D32" target="_self" onmouseover="rs('qc_lang1/D32')" onmouseout="rh()">Def32</a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:69_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> &amp; (  for <font color="Olive" title="b1">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Olive" title="b1">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the    <a href="qc_lang1.html#M2" title="QC_LANG1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#T5" onmouseover="rs('wellord1/T5')" onmouseout="rh()">WELLORD1:5</a>, <a class="ref" href="xboole_1.html#T33" onmouseover="rs('xboole_1/T33')" onmouseout="rh()">XBOOLE_1:33</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_1_1_1_2"><span class="lab"><font color="Green" title="E46">A3</font></span></a></span>;<br/><a NAME="E8:69_1_1_1_2"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
(  not <font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_1_1_1_2"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E9:69_1_1_1_2"/>
 for <font color="Olive" title="b1">s</font> being   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Maroon" title="c6">y</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">s</font> be   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</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"> ( <font color="Maroon" title="c7">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c6">y</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_1_1_1_2_1"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<font color="Maroon" title="c7">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">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"> <font color="Maroon" title="c6">y</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:69_1_1_1_2_1_1"/>
( <font color="Maroon" title="c7">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  or  not <font color="Maroon" title="c7">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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="c6">y</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:69_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:69_1_1_1_2"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="qc_lang1.html#D33" target="_self" onmouseover="rs('qc_lang1/D33')" onmouseout="rh()">Def33</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
 not <font color="Maroon" title="c7">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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="c6">y</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font></span><br/><div class="add"><a NAME="E2:69_1_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1_1_2_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:69_1_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the    <a href="qc_lang1.html#M2" title="QC_LANG1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_1_1_1_2"><span class="lab"><font color="Green" title="E46">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:69_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:69_1_1_1_2"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:69_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="ref" href="qc_lang1.html#D33" target="_self" onmouseover="rs('qc_lang1/D33')" onmouseout="rh()">Def33</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E10:69_1_1_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; (  for <font color="Olive" title="b2">t</font> being   <a href="qc_lang1.html#NM2" title="QC_LANG1:NM.2">QC-symbol</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="qc_lang1.html#R1" title="QC_LANG1:pred.1">&lt;=</a> <font color="Olive" title="b2">t</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:69_1_1_1_2"><span class="lab"><font color="Green" title="E47">A4</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
