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<b>set </b><font color="Maroon" title="c2">M</font> =  <a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font>;<br/>
<a NAME="E1:33_1_1"/><i><font color="Green" title="E15">00</font></i>: 
 <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">P</font></span>)</span> : for <font color="Olive" title="b2">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> holds <br/> ex <font color="Olive" title="b3">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="matroid0.html#D8" target="_self" title="MATROID0:def.8">Def4</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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 <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="matroid0.html#D4" target="_self" title="MATROID0:def.4">MATROID0:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  ex <font color="Olive" title="b1">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> ) )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon" title="c1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E2:33_1_1"/><i><font color="Green" title="E16">0A</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  ex <font color="Olive" title="b1">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> ) )</span><br/>

<a NAME="E3:33_1_1"/><i><font color="Green" title="E17">07</font></i>: 
( <font color="Maroon" title="c3">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> &amp; <font color="Maroon" title="c4">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">independent</a> )
 
<b>by </b><i><a class="txt" href="#E2:33_1_1"><i><font color="Green" title="E16">0A</font></i></a>, <a class="ref" href="matroid0.html#D2" target="_self" title="MATROID0:def.2">Def8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">P</font><b> such that </b><br/><a NAME="E5:33_1_1"/><i><font color="Green" title="E18">06</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 <b>by </b><i><a class="ref" href="matroid0.html#T9" target="_self" title="MATROID0:th.9">Th14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c1">P</font><b> such that </b><br/><a NAME="E7:33_1_1"/><i><font color="Green" title="E19">03</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 <b>by </b><i><a class="ref" href="matroid0.html#T9" target="_self" title="MATROID0:th.9">Th14</a></i>;<br/>
<a NAME="E8:33_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="ref" href="matroid0.html#D8" target="_self" title="MATROID0:def.8">Def4</a></i>;<br/>
<a NAME="E9:33_1_1"/><b>then </b>
( ( <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/>
<a NAME="E10:33_1_1"/><b>then </b><i><font color="Green" title="E20">02</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c5">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c6">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> )
 
<b>by </b><i><a class="txt" href="#E5:33_1_1"><i><font color="Green" title="E18">06</font></i></a>, <a class="txt" href="#E3:33_1_1"><i><font color="Green" title="E17">07</font></i></a>, <a class="txt" href="#E7:33_1_1"><i><font color="Green" title="E19">03</font></i></a>, <a class="ref" href="matroid0.html#T10" target="_self" title="MATROID0:th.10">Th13</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">A'</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c8">B'</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">g</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E12:33_1_1"/>
( <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c7">A'</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c8">B'</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a>  )
 
<b>by </b><i><a class="txt" href="#E10:33_1_1"><i><font color="Green" title="E20">02</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E13:33_1_1"/><b>then </b><i><font color="Green" title="E21">0A</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">A'</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">B'</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>
<b>assume </b><a NAME="E14:33_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> )</span><br/>

<a NAME="E15:33_1_1"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">B</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E17:33_1_1"/><i><font color="Green" title="E22">08</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">A'</font>
 <b>by </b><i><a class="txt" href="#E13:33_1_1"><i><font color="Green" title="E21">0A</font></i></a>, <a class="ref" href="matroid0.html#T5" target="_self" title="MATROID0:th.5">Th11</a></i>;<br/>
<a NAME="E18:33_1_1"/><i><font color="Green" title="E23">05</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B'</font> &amp; <font color="Maroon" title="c9">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c7">A'</font> )
 
<b>by </b><i><a class="txt" href="#E17:33_1_1"><i><font color="Green" title="E22">08</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E20:33_1_1"/><i><font color="Green" title="E24">09</font></i>: 
( <font color="Maroon" title="c10">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x'</font> )
 <b>by </b><i><a class="txt" href="#E10:33_1_1"><i><font color="Green" title="E20">02</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c10">x'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> <b>by </b><i><a class="txt" href="#E20:33_1_1"><i><font color="Green" title="E24">09</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:33_1_1_1"/><b>then </b><i><font color="Green" title="E25">0B</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">g</font> )
 <b>by </b><i><a class="txt" href="#E10:33_1_1"><i><font color="Green" title="E20">02</font></i></a>, <a class="txt" href="#E5:33_1_1"><i><font color="Green" title="E18">06</font></i></a>, <a class="txt" href="#E7:33_1_1"><i><font color="Green" title="E19">03</font></i></a>, <a class="txt" href="#E20:33_1_1"><i><font color="Green" title="E24">09</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:33_1_1_1"/><b>then </b>
( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:33_1_1_1"><i><font color="Green" title="E25">0B</font></i></a>, <a class="txt" href="#E20:33_1_1"><i><font color="Green" title="E24">09</font></i></a>, <a class="txt" href="#E18:33_1_1"><i><font color="Green" title="E23">05</font></i></a>, <a class="ref" href="taxonom2.html#D5" title="TAXONOM2:def.5">TAXONOM2:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E23:33_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="#E20:33_1_1"><i><font color="Green" title="E24">09</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c12">xx</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span> <b>by </b><i><a class="txt" href="#E20:33_1_1"><i><font color="Green" title="E24">09</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">Ax</font> = <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">xx</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">P</font></span>)</span> <b>by </b><i><a class="ref" href="matroid0.html#D8" target="_self" title="MATROID0:def.8">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">D</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="c14">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> implies  ex <font color="Olive" title="b1">x'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c13">Ax</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b1">x'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><b>assume </b><a NAME="E1:33_1_1_2"/><i><font color="Green" title="E25">0C</font></i>: 
<font color="Maroon" title="c14">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c13">Ax</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b1">x'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><b>then consider </b><font color="Maroon" title="c15">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">D</font><b> such that </b><br/><a NAME="E3:33_1_1_2"/><i><font color="Green" title="E26">0D</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E3:33_1_1"><i><font color="Green" title="E17">07</font></i></a>, <a class="ref" href="matroid0.html#T8" target="_self" title="MATROID0:th.8">Th10</a></i>;<br/><a NAME="E4:33_1_1_2"/><i><font color="Green" title="E27">0E</font></i>: 
<font color="Maroon" title="c13">Ax</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">xx</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_2_1"/>
( <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> or <font color="Maroon" title="c14">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_1"/><i><font color="Green" title="E28">0F</font></i>: 
<font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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" title="b1">x'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c13">Ax</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b1">x'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><b>take </b><font color="Maroon" title="c16">x'</font> = <font color="Maroon" title="c10">x'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">Ax</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">x'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><a NAME="E2:33_1_1_2_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">z</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="tarski.html#D3" title="TARSKI:def.3">TARSKI: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="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font> or <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>assume </b><a NAME="E1:33_1_1_2_1_1_1"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E2:33_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E29">0G</font></i>: 
( <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:33_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E30">0H</font></i>: 
( <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font> )
 
<b>by </b><i><a class="txt" href="#E5:33_1_1"><i><font color="Green" title="E18">06</font></i></a>, <a class="txt" href="#E10:33_1_1"><i><font color="Green" title="E20">02</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:33_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c14">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> )
 
<b>by </b><i><a class="txt" href="#E2:33_1_1_2_1_1_1"><i><font color="Green" title="E29">0G</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E5:33_1_1_2_1_1_1"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E1:33_1_1_2_1_1"><i><font color="Green" title="E28">0F</font></i></a>, <a class="txt" href="#E3:33_1_1_2_1_1_1"><i><font color="Green" title="E30">0H</font></i></a>, <a class="txt" href="#E18:33_1_1"><i><font color="Green" title="E23">05</font></i></a>, <a class="ref" href="taxonom2.html#D5" title="TAXONOM2:def.5">TAXONOM2:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:33_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>hence </b><a NAME="E4:33_1_1_2_1_1"/>
<font color="Maroon" title="c13">Ax</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">x'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E4:33_1_1_2"><i><font color="Green" title="E27">0E</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_2"/><i><font color="Green" title="E28">0I</font></i>: 
<font color="Maroon" title="c14">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">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" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c13">Ax</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b1">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c16">d</font> = <font color="Maroon" title="c15">d</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c17">d</font> = <font color="Maroon" title="c16">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">Ax</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><a NAME="E3:33_1_1_2_1_2"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">g</font>
 <b>by </b><i><a class="txt" href="#E20:33_1_1"><i><font color="Green" title="E24">09</font></i></a>, <a class="txt" href="#E10:33_1_1"><i><font color="Green" title="E20">02</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:33_1_1_2_1_2"/><b>then </b>
( <font color="Maroon" title="c9">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c14">D</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">a</font> )
 <b>by </b><i><a class="txt" href="#E7:33_1_1"><i><font color="Green" title="E19">03</font></i></a>, <a class="txt" href="#E20:33_1_1"><i><font color="Green" title="E24">09</font></i></a>, <a class="txt" href="#E1:33_1_1_2_1_2"><i><font color="Green" title="E28">0I</font></i></a>, <a class="txt" href="#E1:33_1_1_2"><i><font color="Green" title="E25">0C</font></i></a>, <a class="ref" href="taxonom2.html#D5" title="TAXONOM2:def.5">TAXONOM2:def 5</a></i>;<br/><a NAME="E5:33_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c14">D</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E6:33_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c12">xx</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a> <font color="Maroon" title="c14">D</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E7:33_1_1_2_1_2"/><b>then </b>
( <font color="Maroon" title="c12">xx</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">xx</font> &amp; <font color="Maroon" title="c12">xx</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">xx</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E8:33_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c12">xx</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/><b>hence </b><a NAME="E9:33_1_1_2_1_2"/>
<font color="Maroon" title="c13">Ax</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c14">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E3:33_1_1_2"><i><font color="Green" title="E26">0D</font></i></a>, <a class="txt" href="#E4:33_1_1_2"><i><font color="Green" title="E27">0E</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E27:33_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <span class="p1">(<span class="default"><a href="matroid0.html#K2" title="MATROID0:func.2">ProdMatroid</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:33_1_1"><i><font color="Green" title="E15">00</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
