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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>   <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>, <font color="Olive" title="b3">b</font> being   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b3">b</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Olive" title="b2">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Olive" title="b1">t</font> holds <br/>( <font color="Olive" title="b3">b</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b2">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Olive" title="b3">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Olive" title="b1">t</font> &amp; <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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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="b3">b</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#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Olive" title="b1">t</font></span>)</span> )</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>, <font color="Olive" title="b2">b</font> being   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c2">t</font> &amp; <font color="Olive" title="b2">b</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Olive" title="b1">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font> holds <br/>( <font color="Olive" title="b2">b</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c2">t</font> &amp; <font color="Olive" title="b1">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Olive" title="b2">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font> &amp; <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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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#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#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</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"> ( <font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c2">t</font> &amp; <font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font> implies ( <font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c2">t</font> &amp; <font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font> &amp; <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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:55"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c2">t</font>
 <span class="kw">and </span><br/><a NAME="E2:55"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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"> ( <font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c2">t</font> &amp; <font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font> &amp; <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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">t9</font> being   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E4:55"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c5">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="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E5:55"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c5">t9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E28">A2</font></span></a></span>;<br/>
<a NAME="E6:55"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<font color="Maroon" title="c4">b</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="c5">t9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="abcmiz_0.html#T13" target="_self" onmouseover="rs('abcmiz_0/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<a NAME="E7:55"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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="#E1:55"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="abcmiz_0.html#T20" target="_self" onmouseover="rs('abcmiz_0/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:55"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c2">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E4:55"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="yellow_0.html#D2" onmouseover="rs('yellow_0/D2')" onmouseout="rh()">YELLOW_0:def 2</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"> ( <font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font> &amp; <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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> )</span><br/>

<a NAME="E9:55"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
<font color="Maroon" title="c5">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="#E7:55"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E30">A4</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/>
<span class="kw">thus </span><a NAME="E10:55"/><span class="lab"><font color="Green" title="E35">A9</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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"> <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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="55_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c5">t9</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abcmiz_0.html#D15" target="_self" onmouseover="rs('abcmiz_0/D15')" onmouseout="rh()">ABCMIZ_0:def 15</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="c5">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> &amp; <font color="Maroon" title="c5">t9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font> )</span><br/>

<a NAME="E1:55_1"/>
<font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</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="#E1:55"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="abcmiz_0.html#T22" target="_self" onmouseover="rs('abcmiz_0/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:55_1"/>
<font color="Maroon" title="c5">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="#E5:55"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="waybel_0.html#D19" onmouseover="rs('waybel_0/D19')" onmouseout="rh()">WAYBEL_0:def 19</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"> <font color="Maroon" title="c5">t9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span><br/>

<span class="kw">thus </span><a NAME="E3:55_1"/>
<font color="Maroon" title="c5">t9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:55"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="abcmiz_0.html#T23" target="_self" onmouseover="rs('abcmiz_0/T23')" onmouseout="rh()">Th23</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>

<a NAME="E11:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A10</font></span>: 
<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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#T20" target="_self" onmouseover="rs('abcmiz_0/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E12:55"/><span class="lab"><font color="Green" title="E37">A11</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</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="#E1:55"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="abcmiz_0.html#T22" target="_self" onmouseover="rs('abcmiz_0/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<a NAME="E13:55"/><span class="lab"><font color="Green" title="E38">A12</font></span>: 
<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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c4">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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="55_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">t9</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice3.html#D9" onmouseover="rs('lattice3/D9')" onmouseout="rh()">LATTICE3:def 9</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">t9</font> <a href="tarski.html#R2" title="TARSKI: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="c4">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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> or <font color="Maroon" title="c6">t9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:55_2"/><span class="lab"><font color="Green" title="E39">A13</font></span>: 
<font color="Maroon" title="c6">t9</font> <a href="tarski.html#R2" title="TARSKI: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="c4">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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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">t9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span><br/>

<a NAME="E2:55_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">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="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:55_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A14</font></span>: 
<font color="Maroon" title="c4">b</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="c6">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="E4:55_2"/>
<font color="Maroon" title="c6">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="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_2"><span class="lab"><font color="Green" title="E39">A13</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="E5:55_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A15</font></span>: 
<font color="Maroon" title="c6">t9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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="E6:55_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">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="#E12:55"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="ref" href="waybel_0.html#D19" onmouseover="rs('waybel_0/D19')" onmouseout="rh()">WAYBEL_0:def 19</a></span>;<br/>
<a NAME="E7:55_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A16</font></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="c6">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="E8:55_2"/>
<font color="Maroon" title="c6">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="#E7:55"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E5:55_2"><span class="lab"><font color="Green" title="E41">A15</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="E9:55_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">t9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55_2"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="ref" href="abcmiz_0.html#T23" target="_self" onmouseover="rs('abcmiz_0/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:55_2"/>
<font color="Maroon" title="c6">t9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:55_2"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="ref" href="abcmiz_0.html#T23" target="_self" onmouseover="rs('abcmiz_0/T23')" onmouseout="rh()">Th23</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>
<a NAME="E14:55"/>
<font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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="#E8:55"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="abcmiz_0.html#T20" target="_self" onmouseover="rs('abcmiz_0/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E15:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A17</font></span>: 
<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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <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="#E11:55"><span class="lab"><font color="Green" title="E36">A10</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="E16:55"/>
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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="#E10:55"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="abcmiz_0.html#T21" target="_self" onmouseover="rs('abcmiz_0/T21')" onmouseout="rh()">Th21</a></span>;<br/>
<a NAME="E17:55"/><span class="kw">then </span>
<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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:55"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="ref" href="abcmiz_0.html#T23" target="_self" onmouseover="rs('abcmiz_0/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<a NAME="E18:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A18</font></span>: 
<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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <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="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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="E19:55"/><span class="lab"><font color="Green" title="E41">A19</font></span>: 
<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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:55"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="abcmiz_0.html#T20" target="_self" onmouseover="rs('abcmiz_0/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E20:55"/>
<font color="Maroon" title="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</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="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:55"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="abcmiz_0.html#T22" target="_self" onmouseover="rs('abcmiz_0/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<a NAME="E21:55"/><span class="kw">then </span>
<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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <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="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:55"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="ref" href="waybel_0.html#D19" onmouseover="rs('waybel_0/D19')" onmouseout="rh()">WAYBEL_0:def 19</a></span>;<br/>
<a NAME="E22:55"/><span class="kw">then </span>
<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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="tarski.html#R2" title="TARSKI: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="c4">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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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="#E18:55"><span class="lab"><font color="Green" title="E40">A18</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="E23:55"/><span class="kw">then </span>
 for <font color="Olive" title="b1">t9</font> being   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">t9</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c4">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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> 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="Maroon" title="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">t9</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E24:55"/>
<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">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">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="c4">b</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c2">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:55"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="ref" href="yellow_0.html#T30" onmouseover="rs('yellow_0/T30')" onmouseout="rh()">YELLOW_0:30</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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