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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>   <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> holds   <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Olive" title="b1">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c2">p</font></span><br/>



<span class="kw">thus </span><a NAME="E1:105"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c2">p</font>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">p</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">p</font></span>)</span> or <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c2">p</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:105_1"/><span class="lab"><font color="Green" title="E70">A1</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">p</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c2">p</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">q</font> = <font color="Maroon" title="c3">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">S</font> =  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">p</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">u</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="E5:105_1"/><span class="lab"><font color="Green" title="E71">A2</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">and </span><br/><a NAME="E6:105_1"/><span class="lab"><font color="Green" title="E72">A3</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_1"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="ref" href="waybel11.html#T43" target="_self" title="WAYBEL11:th.43">Th43</a>, <a class="ref" href="tops_1.html#T51" title="TOPS_1:th.51">TOPS_1:51</a></span>;<br/>
<a NAME="E7:105_1"/>
<font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></span>;<br/>
<a NAME="E8:105_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:105_1"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</a></span>;<br/>
<a NAME="E9:105_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:105_1"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:105_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E2:105"/>
 <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T11" title="WAYBEL_3:th.11">WAYBEL_3:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:105"/>
 <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#T36" target="_self" title="WAYBEL11:th.36">Th36</a>, <a class="ref" href="tops_1.html#T56" title="TOPS_1:th.56">TOPS_1:56</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
