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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">P</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="proofs_1.html#L1" title="PROOFS_1:struct.1">ProofSystem</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">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</font><br/>  for <font color="Olive" title="b2">F</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b1">B</font> is <font color="Olive" title="b2">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a>  holds <br/> ex <font color="Olive" title="b3">B1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B1</font> &amp; <font color="Olive" title="b3">B1</font> is <font color="Olive" title="b2">F</font> <a href="proofs_1.html#V14" title="PROOFS_1:attr.14">-maximally-omitting</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</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">F</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</font>  st <font color="Maroon" title="c2">B</font> is <font color="Olive" title="b1">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a>  holds <br/> ex <font color="Olive" title="b2">B1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</font> st <br/>( <font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B1</font> &amp; <font color="Olive" title="b2">B1</font> is <font color="Olive" title="b1">F</font> <a href="proofs_1.html#V14" title="PROOFS_1:attr.14">-maximally-omitting</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</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="c2">B</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a>  implies  ex <font color="Olive" title="b1">B1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</font> st <br/>( <font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B1</font> &amp; <font color="Olive" title="b1">B1</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V14" title="PROOFS_1:attr.14">-maximally-omitting</a>  ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:93"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon" title="c2">B</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</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">  ex <font color="Olive" title="b1">B1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</font> st <br/>( <font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B1</font> &amp; <font color="Olive" title="b1">B1</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V14" title="PROOFS_1:attr.14">-maximally-omitting</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">A</font> =  the <a href="proofs_1.html#U1" title="PROOFS_1:sel.1">Axioms</a> of <font color="Maroon" title="c1">P</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">R</font> =  the <a href="proofs_1.html#U2" title="PROOFS_1:sel.2">Rules</a> of <font color="Maroon" title="c1">P</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">Q</font> = <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">G</font> = <font color="Maroon" title="c3">F</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">Om</font> =  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</font>);<br/>
<a NAME="E3:93"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
 for <font color="Olive" title="b1">C1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span><br/>  for <font color="Olive" title="b2">B1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b2">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font> holds <br/>( <font color="Olive" title="b1">C1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</font>) iff <font color="Olive" title="b2">B1</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="93_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">C1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</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">  for <font color="Olive" title="b1">B1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font> holds <br/>( <font color="Maroon" title="c9">C1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</font>) iff <font color="Olive" title="b1">B1</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c10">B1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</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="c10">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font> implies ( <font color="Maroon" title="c9">C1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</font>) iff <font color="Maroon" title="c10">B1</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:93_1"/>
<font color="Maroon" title="c10">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</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="c9">C1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</font>) iff <font color="Maroon" title="c10">B1</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a>  )</span><br/>

<a NAME="E2:93_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A3</font></span>: 
 the <a href="proofs_1.html#U1" title="PROOFS_1:sel.1">Axioms</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c9">C1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="proofs_1.html#U1" title="PROOFS_1:sel.1">Axioms</a> of <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K8" title="PROOFS_1:func.8">\/</a> <font color="Maroon" title="c10">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" onmouseover="rs('xboole_1/T4')" onmouseout="rh()">XBOOLE_1:4</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:93_1"/>
( <font color="Maroon" title="c9">C1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</font>) implies <font color="Maroon" title="c10">B1</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</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"> ( <font color="Maroon" title="c10">B1</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a>  implies <font color="Maroon" title="c9">C1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</font>) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="93_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:93_1_1"/>
<font color="Maroon" title="c9">C1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</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="c10">B1</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a> </span><br/>

<a NAME="E2:93_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">C1</font> is <font color="Maroon" title="c7">G</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="proofs_1.html#D25" target="_self" onmouseover="rs('proofs_1/D25')" onmouseout="rh()">Def17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:93_1_1"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> &amp;  not <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c9">C1</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c11">a</font> )
 ;<br/>
<span class="kw">thus </span><a NAME="E5:93_1_1"/>
<font color="Maroon" title="c10">B1</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E4:93_1_1"><span class="lab"><font color="Green" title="E40">A4</font></span></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/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E4:93_1"/>
<font color="Maroon" title="c10">B1</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</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"> <font color="Maroon" title="c9">C1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:93_1"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> &amp;  not <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c10">B1</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c11">a</font> )
 ;<br/>
<a NAME="E7:93_1"/>
<font color="Maroon" title="c9">C1</font> is <font color="Maroon" title="c7">G</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E6:93_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:93_1"/>
<font color="Maroon" title="c9">C1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:93"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
 not  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="93_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c9">E</font> =  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>;<br/>
<a NAME="E1:93_2"/>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">B</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E2:93_2"/>
 not  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:93"><span class="lab"><font color="Green" title="E38">A2</font></span></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/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">Y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</font>)<span class="kw"> such that </span><br/><a NAME="E6:93"/><span class="lab"><font color="Green" title="E40">A11</font></span>: 
 for <font color="Olive" title="b1">Z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</font>)  holds  not <font color="Maroon" title="c9">Y</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b1">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="proofs_1.html#T21" target="_self" onmouseover="rs('proofs_1/T21')" onmouseout="rh()">Th60</a></span>;<br/>
<a NAME="E7:93"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
<font color="Maroon" title="c9">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:93"><span class="lab"><font color="Green" title="E39">A10</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">B1</font> = <font color="Maroon" title="c2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">Y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">B1</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="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B1</font> &amp; <font color="Maroon" title="c10">B1</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V14" title="PROOFS_1:attr.14">-maximally-omitting</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E9:93"/>
<font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</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"> <font color="Maroon" title="c10">B1</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V14" title="PROOFS_1:attr.14">-maximally-omitting</a> </span><br/>

<span class="kw">thus </span><a NAME="E10:93"/>
<font color="Maroon" title="c10">B1</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E7:93"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="proofs_1.html#D26" target="_self" onmouseover="rs('proofs_1/D26')" onmouseout="rh()">PROOFS_1:def 26</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">B2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</font>  st <font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b1">B2</font> holds <br/> not <font color="Olive" title="b1">B2</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">B2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</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="c10">B1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c11">B2</font> implies  not <font color="Maroon" title="c11">B2</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">C2</font> = <font color="Maroon" title="c11">B2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span> ;<br/>
<span class="kw">assume </span><a NAME="E12:93"/><span class="lab"><font color="Green" title="E42">A15</font></span>: 
<font color="Maroon" title="c10">B1</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c11">B2</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">  not <font color="Maroon" title="c11">B2</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a> </span><br/>

<span class="kw">assume </span><a NAME="E13:93"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<font color="Maroon" title="c11">B2</font> is <font color="Maroon" title="c3">F</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</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"> contradiction</span><br/>

<a NAME="E14:93"/>
<font color="Maroon" title="c2">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c11">B2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:93"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T59" onmouseover="rs('xboole_1/T59')" onmouseout="rh()">XBOOLE_1:59</a></span>;<br/>
<a NAME="E15:93"/><span class="kw">then </span>
<font color="Maroon" title="c11">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>;<br/>
<a NAME="E16:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A18</font></span>: 
<font color="Maroon" title="c12">C2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="proofs_1.html#K10" title="PROOFS_1:func.10">Omit</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c7">G</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E13:93"><span class="lab"><font color="Green" title="E43">A16</font></span></a></span>;<br/>
<a NAME="E17:93"/>
<font color="Maroon" title="c9">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:93"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:93"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E12:93"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E16:93"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="ref" href="xboole_1.html#T59" onmouseover="rs('xboole_1/T59')" onmouseout="rh()">XBOOLE_1:59</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/>


</div>
