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<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">f</font><br/> for <font color="Olive" title="b3">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Olive" title="b1">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">t</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b3">A</font></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">f</font></span><br/><b>let </b><font color="Maroon" title="c2">s</font>, <font color="Maroon" title="c3">t</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">f</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon" title="c2">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">A</font></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">f</font></span><br/><b>let </b><font color="Maroon" title="c4">A</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="c2">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c4">A</font></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">f</font></span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1"/>
( <font color="Maroon" title="c1">f</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> or  not <font color="Maroon" title="c1">f</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1"><b>suppose </b></a><a NAME="E1:108_1_1"/>
<font color="Maroon" title="c1">f</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c4">A</font></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">f</font></span><br/><div class="add"><a NAME="E2:108_1_1"/><b>then </b>
not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font>
 ;<br/><a NAME="E3:108_1_1"/><b>then </b>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">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> 
 <b>by </b><i><a class="ref" href="card_3.html#T37" target="_self" title="CARD_3:th.37">Th37</a></i>;<br/><a NAME="E4:108_1_1"/><b>then </b><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">f</font>
 ;<br/><a NAME="E5:108_1_1"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="ref" href="card_3.html#T67" target="_self" title="CARD_3:th.67">Th67</a></i>;<br/><b>hence </b><a NAME="E6:108_1_1"/>
<font color="Maroon" title="c2">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c4">A</font></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="#E4:108_1_1"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="card_3.html#T69" target="_self" title="CARD_3:th.69">Th69</a>, <a class="ref" href="card_3.html#T81" target="_self" title="CARD_3:th.81">Th81</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="108_1_2"><b>suppose </b></a><a NAME="E1:108_1_2"/>
not <font color="Maroon" title="c1">f</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c4">A</font></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">f</font></span><br/><div class="add"><a NAME="E2:108_1_2"/><b>then </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D9" title="RELAT_1:def.9">RELAT_1:def 9</a></i>;<br/><a NAME="E3:108_1_2"/><b>then </b><i><font color="Green" title="E54">A</font></i>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">f</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="card_3.html#T37" target="_self" title="CARD_3:th.37">Th37</a></i>;<br/><a NAME="E4:108_1_2"/><b>then </b><i><font color="Green" title="E55">B</font></i>: 
<font color="Maroon" title="c2">s</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="subset_1.html#D2" title="SUBSET_1:def.2">SUBSET_1:def 2</a></i>;<br/><a NAME="E5:108_1_2"/>
<font color="Maroon" title="c3">t</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="txt" href="#E3:108_1_2"><i><font color="Green" title="E54">A</font></i></a>, <a class="ref" href="subset_1.html#D2" title="SUBSET_1:def.2">SUBSET_1:def 2</a></i>;<br/><a NAME="E6:108_1_2"/><b>then </b>
<font color="Maroon" title="c3">t</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c4">A</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="E7:108_1_2"/>
<font color="Maroon" title="c2">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c4">A</font></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="#E4:108_1_2"><i><font color="Green" title="E55">B</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>

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