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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <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="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds   <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">f</font>)</span>)</span>, <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b1">f</font></span>)</span>) <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font>)</span>)</span>, <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font></span>)</span>) <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> </span><br/>



<a NAME="E1:63"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T10" onmouseover="rs('card_3/T10')" onmouseout="rh()">CARD_3:10</a>, <a class="ref" href="funct_5.html#T59" onmouseover="rs('funct_5/T59')" onmouseout="rh()">FUNCT_5:59</a></span>;<br/>
<span class="lab"><font color="Green" title="E34">A2</font></span>:  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c2">f</font>)</span>)</span> = 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#T50" target="_self" onmouseover="rs('funct_6/T50')" onmouseout="rh()">Th45</a></span>
<br/>.= 
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T11" onmouseover="rs('card_3/T11')" onmouseout="rh()">CARD_3:11</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_5.html#T59" onmouseover="rs('funct_5/T59')" onmouseout="rh()">FUNCT_5:59</a></span>
;<br/>
<a NAME="E3:63"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
(  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c1">X</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#T51" target="_self" onmouseover="rs('funct_6/T51')" onmouseout="rh()">Th46</a>, <a class="ref" href="card_3.html#T10" onmouseover="rs('card_3/T10')" onmouseout="rh()">CARD_3:10</a>, <a class="ref" href="funct_5.html#T57" onmouseover="rs('funct_5/T57')" onmouseout="rh()">FUNCT_5:57</a></span>;<br/>
<a NAME="E4:63"/>
( <font color="Maroon" title="c1">X</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>  &amp; <font color="Maroon" title="c2">f</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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font>)</span>)</span>, <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font></span>)</span>) <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="funct_6.html#E62"><span class="lab"><font color="Green" title="E30">Lm4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:63"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font>)</span>)</span>, <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font></span>)</span>) <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="card_1.html#T28" onmouseover="rs('card_1/T28')" onmouseout="rh()">CARD_1:28</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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