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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</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"> ( <font color="Maroon" title="c1">A</font> <a href="setfam_1.html#R2" title="SETFAM_1:pred.2">is_coarser_than</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c2">B</font> <a href="setfam_1.html#R2" title="SETFAM_1:pred.2">is_coarser_than</a> <font color="Maroon" title="c3">C</font> implies <font color="Maroon" title="c1">A</font> <a href="setfam_1.html#R2" title="SETFAM_1:pred.2">is_coarser_than</a> <font color="Maroon" title="c3">C</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="setfam_1.html#R2" title="SETFAM_1:pred.2">is_coarser_than</a> <font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c2">B</font> <a href="setfam_1.html#R2" title="SETFAM_1:pred.2">is_coarser_than</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">A</font> <a href="setfam_1.html#R2" title="SETFAM_1:pred.2">is_coarser_than</a> <font color="Maroon" title="c3">C</font></span><br/>

<b>let </b><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="setfam_1.html#D3" title="SETFAM_1:def.3">SETFAM_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font> ) )</span><br/>

<b>assume </b><a NAME="E3:4"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c5">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:4"/><i><font color="Green" title="E5">A3</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E6:4"/><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="setfam_1.html#D3" title="SETFAM_1:def.3">SETFAM_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:4"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <b>and </b><br/><a NAME="E9:4"/><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="#E2:4"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="#E5:4"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="setfam_1.html#D3" title="SETFAM_1:def.3">SETFAM_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c6">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font> )</span><br/>

<b>thus </b><a NAME="E10:4"/>
( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c6">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font> )
 <b>by </b><i><a class="txt" href="#E6:4"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="#E8:4"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="#E9:4"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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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