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<span class="kw">let </span><font color="Maroon" title="c2">S1</font>, <font color="Maroon" title="c3">S2</font> be   <a href="fintopo7.html#NM3" title="FINTOPO7:NM.3">FMT_TopSpace</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="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c3">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> implies <font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45_1_1"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> )
 <span class="kw">and </span><br/><a NAME="E2:45_1_1"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp;  <a href="fintopo7.html#K4" title="FINTOPO7:func.4">Family_open_set</a> <font color="Maroon" title="c3">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</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="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">F1</font> =  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c2">S1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">F2</font> =  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c3">S2</font>;<br/>
<a NAME="E3:45_1_1"/>
 the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c2">S1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c3">S2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c6">F2</font> =  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c3">S2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S1</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S1</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/>
<a NAME="E2:45_1_1_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</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">S1</font> holds <br/> the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c2">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c7">x</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">S1</font> implies  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c2">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_1_1_1_1"/>
<font color="Maroon" title="c7">x</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">S1</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">  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c2">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font> ;<br/>
<a NAME="E3:45_1_1_1_1"/>
 the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c2">S1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c2">S1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c9">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c2">S1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> implies <font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:45_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c2">S1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</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="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font></span><br/><a NAME="E2:45_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c8">x</font>
 ;<br/><a NAME="E3:45_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S1</font> :  ex <font color="Olive" title="b2">O</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">O</font> &amp; <font color="Olive" title="b2">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="fintopo7.html#T20" target="_self" onmouseover="rs('fintopo7/T20')" onmouseout="rh()">Th13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S1</font><span class="kw"> such that </span><br/><a NAME="E5:45_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">V1</font>
 <span class="kw">and </span><br/><a NAME="E6:45_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
 ex <font color="Olive" title="b1">O</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp; <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">V1</font> )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c11">O1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E8:45_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
( <font color="Maroon" title="c11">O1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">O1</font> &amp; <font color="Maroon" title="c11">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">V1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">V2</font> = <font color="Maroon" title="c10">V1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">x2</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/><a NAME="E11:45_1_1_1_1_1_1"/>
( <font color="Maroon" title="c11">O1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c13">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">O1</font> &amp; <font color="Maroon" title="c11">O1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">V2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>;<br/><a NAME="E12:45_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</font> :  ex <font color="Olive" title="b2">O</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">O</font> &amp; <font color="Olive" title="b2">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>;<br/><a NAME="E13:45_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c13">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="fintopo7.html#T20" target="_self" onmouseover="rs('fintopo7/T20')" onmouseout="rh()">Th13</a></span>;<br/><span class="kw">hence </span><a NAME="E14:45_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</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"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c9">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI: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"> (  not <font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> or <font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c2">S1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:45_1_1_1_1_1"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</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="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c2">S1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">x3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font><span class="kw"> such that </span><br/><a NAME="E4:45_1_1_1_1_1"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/>
<a NAME="E5:45_1_1_1_1_1"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c10">x3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_1"><span class="lab"><font color="Green" title="E25">A6</font></span></a></span>;<br/>
<a NAME="E6:45_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</font> :  ex <font color="Olive" title="b2">O</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c10">x3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">O</font> &amp; <font color="Olive" title="b2">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="fintopo7.html#T20" target="_self" onmouseover="rs('fintopo7/T20')" onmouseout="rh()">Th13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">V2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</font><span class="kw"> such that </span><br/><a NAME="E8:45_1_1_1_1_1"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">V2</font>
 <span class="kw">and </span><br/><a NAME="E9:45_1_1_1_1_1"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
 ex <font color="Olive" title="b1">O</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c10">x3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp; <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">V2</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">O2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E11:45_1_1_1_1_1"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
( <font color="Maroon" title="c12">O2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c10">x3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">O2</font> &amp; <font color="Maroon" title="c12">O2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">V2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">V1</font> = <font color="Maroon" title="c11">V2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c10">x3</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/>
<a NAME="E14:45_1_1_1_1_1"/>
( <font color="Maroon" title="c12">O2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c14">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">O2</font> &amp; <font color="Maroon" title="c12">O2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">V1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_1_1_1_1_1"><span class="lab"><font color="Green" title="E29">A10</font></span></a></span>;<br/>
<a NAME="E15:45_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S1</font> :  ex <font color="Olive" title="b2">O</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">O</font> &amp; <font color="Olive" title="b2">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a></span>;<br/>
<a NAME="E16:45_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo2.html#K6" title="FINTOPO2:func.6">U_FMT</a> <font color="Maroon" title="c14">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E1:45_1_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="fintopo7.html#T20" target="_self" onmouseover="rs('fintopo7/T20')" onmouseout="rh()">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:45_1_1_1_1_1"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c2">S1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A7</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:45_1_1_1_1"/>
 the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c2">S1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:45_1_1_1"/>
 the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c2">S1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  the <a href="fintopo2.html#U1" title="FINTOPO2:sel.1">BNbd</a> of <font color="Maroon" title="c3">S2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</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>
<span class="kw">hence </span><a NAME="E4:45_1_1"/>
<font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</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"> verum</span><br/>


</div>
