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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  for <font color="Olive" title="b1">F</font> being   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">K</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">K</font> holds <br/>(  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_3.html#K10" title="PROB_3:func.10">monotoneclass</a> <font color="Olive" title="b1">F</font> &amp;  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">K</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of <font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">K</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">K</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">K</font> holds <br/>(  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_3.html#K10" title="PROB_3:func.10">monotoneclass</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">K</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font> be   <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" 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="c3">K</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font> implies (  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_3.html#K10" title="PROB_3:func.10">monotoneclass</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:115"/><span class="lab"><font color="Green" title="E80">A1</font></span>: 
<font color="Maroon" title="c3">K</font> is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E2:115"/><span class="lab"><font color="Green" title="E81">A2</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</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"> (  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_3.html#K10" title="PROB_3:func.10">monotoneclass</a> <font color="Maroon" title="c2">F</font> &amp;  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font> )</span><br/>

<span class="kw">thus </span><a NAME="E3:115"/>
 <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_3.html#K10" title="PROB_3:func.10">monotoneclass</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T73" onmouseover="rs('prob_3/T73')" onmouseout="rh()">PROB_3:73</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">  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font></span><br/>

<span class="kw">hence </span><a NAME="E4:115"/>
 <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E80">A1</font></span></a>, <a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E81">A2</font></span></a>, <a class="ref" href="prob_3.html#D11" onmouseover="rs('prob_3/D11')" onmouseout="rh()">PROB_3:def 11</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/>


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