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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span></span><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:73_1_1"/>
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c1">A</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1"><span class="kw">case </span></a><a NAME="E1:73_1_1_1"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span></span><br/><div class="add"><a NAME="E2:73_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font>
 ;<br/><span class="kw">hence </span><a NAME="E3:73_1_1_1"/>
 <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="card_3.html#T57" target="_self" title="CARD_3:th.57">Th73</a>, <a class="ref" href="partfun1.html#T48" title="PARTFUN1:th.48">PARTFUN1:48</a></span>; <a class="txt" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_2"><span class="kw">case </span></a><a NAME="E1:73_1_1_2"/>
<font color="Maroon" title="c1">A</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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span> &amp;  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) )</span><br/><div class="add"><a NAME="E2:73_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A2</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</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"><font color="Maroon" title="c2">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T8" title="FUNCOP_1:th.8">FUNCOP_1:8</a></span>;<br/><a NAME="E3:73_1_1_2"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E4:73_1_1_2"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_1_2"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T25" title="ZFMISC_1:th.25">ZFMISC_1:25</a></span>;<br/><span class="kw">thus </span><a NAME="E5:73_1_1_2"/>
 <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:73_1_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:73_1_1_2_1"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_1_2_1"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:73_1_1_2_1"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D3" title="PARTFUN1:def.3">PARTFUN1:def 3</a></span>;<br/>
<a NAME="E6:73_1_1_2_1"/><span class="lab"><font color="Green" title="E46">A8</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_1_2_1"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> implies <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:73_1_1_2_1_1"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span><br/><a NAME="E2:73_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <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="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E3:73_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_1_2_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:73_1_1_2_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_1_2_1"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E1:73_1_1_2_1_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>; <a class="txt" 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">hence </span><a NAME="E8:73_1_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_2_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E6:73_1_1_2_1"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="ref" href="card_3.html#D9" target="_self" title="CARD_3:def.9">Def9</a></span>; <a class="txt" 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">thus </span><a NAME="E6:73_1_1_2"/>
 <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_2"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E4:73_1_1_2"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="card_3.html#T55" target="_self" title="CARD_3:th.55">Th71</a></span>; <a class="txt" 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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:73"/>
 <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" 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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