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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">F</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#V1" title="INTERVA1:attr.1">ordered</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">G</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/>( <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Olive" title="b1">F</font> iff for <font color="Olive" title="b3">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">G</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#V1" title="INTERVA1:attr.1">ordered</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</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">G</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/>( <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c2">F</font> iff for <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">G</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> implies ( <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c2">F</font> iff for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E28">A</font></span>: 
<font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</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">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c2">F</font> iff for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font> )</span><br/>

<span class="kw">thus </span><a NAME="E2:38"/>
( <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c2">F</font> implies for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>; <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">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font> ) implies <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c2">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:38"/><span class="lab"><font color="Green" title="E29">B1</font></span>: 
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</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">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c2">F</font></span><br/>

<a NAME="E4:38"/><span class="lab"><font color="Green" title="E30">B2</font></span>: 
<font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c2">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E28">A</font></span></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E5:38"/>
 <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E29">B1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:38"/>
<font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E30">B2</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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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