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<b>let </b><font color="Maroon" title="c1">TS</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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">F</font>, <font color="Olive" title="b2">G</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">TS</font>  st <font color="Olive" title="b1">F</font> is <a href="setfam_1.html#V3" title="SETFAM_1:attr.3">with_proper_subsets</a> &amp; <font color="Olive" title="b2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b2">G</font> is <a href="setfam_1.html#V3" title="SETFAM_1:attr.3">with_proper_subsets</a></span><br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">TS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is <a href="setfam_1.html#V3" title="SETFAM_1:attr.3">with_proper_subsets</a> &amp; <font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c3">G</font> is <a href="setfam_1.html#V3" title="SETFAM_1:attr.3">with_proper_subsets</a> )</span><br/>



<b>assume </b><a NAME="E1:67"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c2">F</font> is <a href="setfam_1.html#V3" title="SETFAM_1:attr.3">with_proper_subsets</a> &amp; <font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</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">G</font> is <a href="setfam_1.html#V3" title="SETFAM_1:attr.3">with_proper_subsets</a></span><br/>

<b>assume </b><a NAME="E2:67"/>
not <font color="Maroon" title="c3">G</font> is <a href="setfam_1.html#V3" title="SETFAM_1:attr.3">with_proper_subsets</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:67"/><b>then </b>
<font color="Maroon" title="c1">TS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D13" target="_self" title="SETFAM_1:def.13">Def6</a></i>;<br/>
<b>hence </b><a NAME="E4:67"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:67"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="setfam_1.html#D13" target="_self" title="SETFAM_1:def.13">Def6</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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