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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</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="c1">T</font> is  <a href="abcmiz_0.html#V2" title="ABCMIZ_0:attr.2">Mizar-widening-like</a>  implies ( <font color="Maroon" title="c1">T</font> is  <a href="abcmiz_0.html#V1" title="ABCMIZ_0:attr.1">Noetherian</a>  &amp; <font color="Maroon" title="c1">T</font> is  <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>  &amp; <font color="Maroon" title="c1">T</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7_1_1"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c1">T</font> is   <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>
 <span class="kw">and </span><br/><a NAME="E2:7_1_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="abcmiz_0.html#V1" title="ABCMIZ_0:attr.1">Noetherian</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abcmiz_0.html#D3" target="_self" onmouseover="rs('abcmiz_0/D3')" onmouseout="rh()">ABCMIZ_0: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"> ( <font color="Maroon" title="c1">T</font> is  <a href="abcmiz_0.html#V1" title="ABCMIZ_0:attr.1">Noetherian</a>  &amp; <font color="Maroon" title="c1">T</font> is  <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>  &amp; <font color="Maroon" title="c1">T</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">S</font> = <font color="Maroon" title="c1">T</font> as   <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a></span>;<br/>
<a NAME="E4:7_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E6:7_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">and </span><br/><a NAME="E7:7_1_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
 for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/> not <font color="Maroon" title="c3">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="abcmiz_0.html#D2" target="_self" onmouseover="rs('abcmiz_0/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:7_1_1"/>
( <font color="Maroon" title="c1">T</font> is  <a href="abcmiz_0.html#V1" title="ABCMIZ_0:attr.1">Noetherian</a>  &amp; <font color="Maroon" title="c1">T</font> is  <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E2:7_1_1"><span class="lab"><font color="Green" title="E4">A2</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"> <font color="Maroon" title="c1">T</font> is  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c3">a</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow_0.html#D5" onmouseover="rs('yellow_0/D5')" onmouseout="rh()">YELLOW_0:def 5</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">  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c3">a</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> or <font color="Maroon" title="c4">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> )</span><br/>

<a NAME="E9:7_1_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
;<br/>
<a NAME="E10:7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" onmouseover="rs('yellow_0/T22')" onmouseout="rh()">YELLOW_0:22</a></span>;<br/>
<a NAME="E11:7_1_1"/>
 not <font color="Maroon" title="c3">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c3">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7_1_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E9:7_1_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>;<br/>
<a NAME="E12:7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:7_1_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="orders_2.html#D6" onmouseover="rs('orders_2/D6')" onmouseout="rh()">ORDERS_2:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:7_1_1"/>
(  not <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> or <font color="Maroon" title="c4">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="yellow_0.html#T22" onmouseover="rs('yellow_0/T22')" onmouseout="rh()">YELLOW_0:22</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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