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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>





<span class="kw">consider </span><font color="Maroon" title="c4">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E2:45"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E3:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A1</font></span>: 
 <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T27" target="_self" title="ENS_1:th.27">Th27</a></span>;<br/>
<a NAME="E4:45"/>
 <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T17" target="_self" title="ENS_1:th.17">Th17</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">m</font> = <font color="Maroon" title="c4">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E6:45"/>
<font color="Maroon" title="c5">m</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="ens_1.html#T20" target="_self" title="ENS_1:th.20">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:45"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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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