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<b>let </b><font color="Maroon" title="c3">SF</font>, <font color="Maroon" title="c4">SG</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">N</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> holds <br/> ( <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">SF</font> iff ( <font color="Olive" title="b1">N</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> &amp; <font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">N</font> ) ) ) &amp; ( for <font color="Olive" title="b1">N</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> holds <br/> ( <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">SG</font> iff ( <font color="Olive" title="b1">N</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> &amp; <font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">N</font> ) ) ) implies <font color="Maroon" title="c3">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">SG</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_2"/><i><font color="Green" title="E26">A1</font></i>: 
for <font color="Olive" title="b1">N</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> holds <br/> ( <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">SF</font> iff ( <font color="Olive" title="b1">N</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> &amp; <font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">N</font> ) )
 <b>and </b><br/><a NAME="E2:47_1_2"/><i><font color="Green" title="E27">A2</font></i>: 
for <font color="Olive" title="b1">N</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> holds <br/> ( <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">SG</font> iff ( <font color="Olive" title="b1">N</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> &amp; <font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">N</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">SG</font></span><br/>

<a NAME="E3:47_1_2"/>
for <font color="Olive" title="b1">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> holds <br/> ( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">SF</font> iff <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">SG</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">SF</font> iff <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">SG</font> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">SG</font> implies <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">SF</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:47_1_2_1_1"/>
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">SF</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">SG</font></span><br/><a NAME="E2:47_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c5">Y</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> &amp; <font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> )
 <b>by </b><i><a class="txt" href="#E1:47_1_2"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:47_1_2_1_1"/>
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">SG</font>
 <b>by </b><i><a class="txt" href="#E2:47_1_2"><i><font color="Green" title="E27">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:47_1_2_1"/>
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">SG</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">SF</font></span><br/>

<a NAME="E3:47_1_2_1"/><b>then </b>
( <font color="Maroon" title="c5">Y</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> &amp; <font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> )
 
<b>by </b><i><a class="txt" href="#E2:47_1_2"><i><font color="Green" title="E27">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:47_1_2_1"/>
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">SF</font>
 <b>by </b><i><a class="txt" href="#E1:47_1_2"><i><font color="Green" title="E26">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:47_1_2"/>
<font color="Maroon" title="c3">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">SG</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T44" title="SETFAM_1:th.44">SETFAM_1:44</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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