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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>   <a href="abcmiz_0.html#V1" title="ABCMIZ_0:attr.1">Noetherian</a>   non  <a href="abcmiz_0.html#V4" title="ABCMIZ_0:attr.4">void</a>   <a href="abcmiz_0.html#V9" title="ABCMIZ_0:attr.9">adj-structured</a>   <a href="abcmiz_0.html#L2" title="ABCMIZ_0:struct.2">TA-structure</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">  for <font color="Olive" title="b1">t</font> being   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b2">a</font> being   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b4">B</font> <a href="abcmiz_0.html#R2" title="ABCMIZ_0:pred.2">is_applicable_to</a> <font color="Olive" title="b1">t</font> holds <br/><font color="Olive" title="b2">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Olive" title="b1">t</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">t</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</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">a</font> being   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b3">B</font> <a href="abcmiz_0.html#R2" title="ABCMIZ_0:pred.2">is_applicable_to</a> <font color="Maroon" title="c2">t</font> holds <br/><font color="Olive" title="b1">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</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">A</font>, <font color="Olive" title="b2">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b2">B</font> <a href="abcmiz_0.html#R2" title="ABCMIZ_0:pred.2">is_applicable_to</a> <font color="Maroon" title="c2">t</font> holds <br/><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</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">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c5">B</font> <a href="abcmiz_0.html#R2" title="ABCMIZ_0:pred.2">is_applicable_to</a> <font color="Maroon" title="c2">t</font> implies <font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:88"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E2:88"/><span class="lab"><font color="Green" title="E51">A2</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="abcmiz_0.html#R2" title="ABCMIZ_0:pred.2">is_applicable_to</a> <font color="Maroon" title="c2">t</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="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span><br/>

<a NAME="E3:88"/><span class="lab"><font color="Green" title="E52">A3</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#R2" title="ABCMIZ_0:pred.2">is_applicable_to</a> <font color="Maroon" title="c2">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="abcmiz_0.html#T54" target="_self" onmouseover="rs('abcmiz_0/T54')" onmouseout="rh()">Th54</a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E4:88"/><span class="lab"><font color="Green" title="E53">A4</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E5:88"/><span class="lab"><font color="Green" title="E54">A5</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E6:88"/>
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K4" title="ABCMIZ_0:func.4">types</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">t</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="88_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:88_1"/>
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K4" title="ABCMIZ_0:func.4">types</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">t</font></span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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"> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K4" title="ABCMIZ_0:func.4">types</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="88_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K4" title="ABCMIZ_0:func.4">types</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">t</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:88_1_1"/><span class="lab"><font color="Green" title="E55">A6</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</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"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K4" title="ABCMIZ_0:func.4">types</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">t</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">t9</font> = <font color="Maroon" title="c6">x</font> as   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E3:88_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_1_1"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:88_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c7">t9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#T13" target="_self" onmouseover="rs('abcmiz_0/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<a NAME="E5:88_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A7</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c7">t9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/>
<a NAME="E6:88_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_1_1"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E7:88_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A8</font></span>: 
<font color="Maroon" title="c7">t9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" onmouseover="rs('waybel_0/T17')" onmouseout="rh()">WAYBEL_0:17</a></span>;<br/>
<a NAME="E8:88_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A9</font></span>: 
 <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c7">t9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#T10" target="_self" onmouseover="rs('abcmiz_0/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E9:88_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="ref" href="abcmiz_0.html#T49" target="_self" onmouseover="rs('abcmiz_0/T49')" onmouseout="rh()">Th49</a></span>;<br/>
<a NAME="E10:88_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:88_1_1"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="ref" href="yellow_0.html#D2" onmouseover="rs('yellow_0/D2')" onmouseout="rh()">YELLOW_0:def 2</a></span>;<br/>
<a NAME="E11:88_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A10</font></span>: 
<font color="Maroon" title="c7">t9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" onmouseover="rs('waybel_0/T17')" onmouseout="rh()">WAYBEL_0:17</a></span>;<br/>
<a NAME="E12:88_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="ref" href="abcmiz_0.html#T50" target="_self" onmouseover="rs('abcmiz_0/T50')" onmouseout="rh()">Th50</a></span>;<br/>
<a NAME="E13:88_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c7">t9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:88_1_1"><span class="lab"><font color="Green" title="E58">A9</font></span></a></span>;<br/>
<a NAME="E14:88_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c7">t9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E5:88_1_1"><span class="lab"><font color="Green" title="E56">A7</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/>
<a NAME="E15:88_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="abcmiz_0.html#K4" title="ABCMIZ_0:func.4">types</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#T14" target="_self" onmouseover="rs('abcmiz_0/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:88_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K4" title="ABCMIZ_0:func.4">types</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:88_1_1"><span class="lab"><font color="Green" title="E59">A10</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</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">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K4" title="ABCMIZ_0:func.4">types</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">t</font></span>)</span> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:88_1"/><span class="lab"><font color="Green" title="E55">A11</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K4" title="ABCMIZ_0:func.4">types</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">t</font></span>)</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"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">t9</font> = <font color="Maroon" title="c6">x</font> as   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E4:88_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:88_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A12</font></span>: 
<font color="Maroon" title="c7">t9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" onmouseover="rs('waybel_0/T17')" onmouseout="rh()">WAYBEL_0:17</a></span>;<br/>
<a NAME="E6:88_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_0.html#K4" title="ABCMIZ_0:func.4">types</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E7:88_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A13</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c7">t9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#T14" target="_self" onmouseover="rs('abcmiz_0/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E8:88_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c7">t9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88"><span class="lab"><font color="Green" title="E54">A5</font></span></a></span>;<br/>
<a NAME="E9:88_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88_1"><span class="lab"><font color="Green" title="E56">A12</font></span></a>, <a class="ref" href="abcmiz_0.html#T52" target="_self" onmouseover="rs('abcmiz_0/T52')" onmouseout="rh()">Th52</a></span>;<br/>
<a NAME="E10:88_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A14</font></span>: 
<font color="Maroon" title="c7">t9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" onmouseover="rs('waybel_0/T17')" onmouseout="rh()">WAYBEL_0:17</a></span>;<br/>
<a NAME="E11:88_1"/>
<font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/>
<a NAME="E12:88_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:88_1"><span class="lab"><font color="Green" title="E57">A13</font></span></a>, <a class="ref" href="abcmiz_0.html#T13" target="_self" onmouseover="rs('abcmiz_0/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:88_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:88_1"><span class="lab"><font color="Green" title="E58">A14</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</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="E7:88"/>
<font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</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>
