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<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:7_1_1"/><i><font color="Green" title="E3">A1</font></i>: 
( <font color="Maroon" title="c1">T</font> is   <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a> &amp; <font color="Maroon" title="c1">T</font> is <a href="abcmiz_0.html#V1" title="ABCMIZ_0:attr.1">Noetherian</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D3" target="_self" title="ABCMIZ_0:def.3">ABCMIZ_0:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>hence </b><a NAME="E2:7_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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>reconsider </b><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> <b>by </b><i><a class="txt" href="#E1:7_1_1"><i><font color="Green" title="E3">A1</font></i></a></i>;<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/>
<b>then consider </b><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><b> such that </b><br/><a NAME="E6:7_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN: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>
 <b>and </b><br/><a NAME="E7:7_1_1"/><i><font color="Green" title="E4">A2</font></i>: 
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="hidden.html#R2" title="HIDDEN: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>
 <b>by </b><i><a class="txt" href="#E1:7_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="abcmiz_0.html#D2" target="_self" title="ABCMIZ_0:def.2">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">a</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D5" title="YELLOW_0:def.5">YELLOW_0:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>let </b><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN: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="E8:7_1_1"/>
<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#R2" title="HIDDEN: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="E9:7_1_1"/><b>then </b>
( <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> &amp; 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> )
 
<b>by </b><i><a class="txt" href="#E7:7_1_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/>
<a NAME="E10:7_1_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/>
<b>hence </b><a NAME="E11:7_1_1"/>
(  not <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN: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> )
 <b>by </b><i><a class="txt" href="#E1:7_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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