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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</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">X0</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">X0</font>  st <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b2">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  holds <br/><font color="Olive" title="b3">B</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">X0</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>; <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">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font>  st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  holds <br/><font color="Olive" title="b2">B</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <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">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font>  st <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon" title="c3">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  holds <br/><font color="Olive" title="b1">B</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font>; <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">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c3">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  implies <font color="Maroon" title="c4">B</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:57"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 ; <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">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  or <font color="Maroon" title="c4">B</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">C</font> = <font color="Maroon" title="c3">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:57"/>
<font color="Maroon" title="c3">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 ; <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="c4">B</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> </span><br/>

<a NAME="E4:57"/><span class="kw">then </span>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">A</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_3.html#T35" target="_self" title="TOPS_3:th.35">Th35</a></span>;<br/>
<a NAME="E5:57"/><span class="kw">then </span>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">C</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_3.html#T56" target="_self" title="TOPS_3:th.56">Th56</a>, <a class="ref" href="tops_3.html#T59" target="_self" title="TOPS_3:th.59">Th59</a></span>;<br/>
<a NAME="E6:57"/><span class="kw">then </span>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">B</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="tops_1.html#T48" title="TOPS_1:th.48">TOPS_1:48</a>, <a class="ref" href="tops_1.html#T79" title="TOPS_1:th.79">TOPS_1:79</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:57"/>
<font color="Maroon" title="c4">B</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_3.html#T35" target="_self" title="TOPS_3:th.35">Th35</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/>


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