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<a NAME="E1:73_1_1"/><i><font color="Green" title="E41">A2</font></i>: 
( ( <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m1</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m1</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <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="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m1</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> ,<span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m1</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m1</font></span>)</span> <a href="coh_sp.html#K19" title="COH_SP:func.19">`1</a>  holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m1</font></span>)</span> <a href="coh_sp.html#K19" title="COH_SP:func.19">`1</a>  ) &amp; ( <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <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="c3">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> ,<span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="coh_sp.html#K19" title="COH_SP:func.19">`1</a>  holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="coh_sp.html#K19" title="COH_SP:func.19">`1</a>  ) )
 
<b>by </b><i><a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th44</a></i>;<br/>
<a NAME="E2:73_1_1"/><b>then </b><i><font color="Green" title="E42">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m1</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> ,<span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> 
 
<b>by </b><i><a class="txt" href="coh_sp.html#E1:73"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:73_1_1_1"/><i><font color="Green" title="E43">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m1</font></span>)</span> <a href="coh_sp.html#K19" title="COH_SP:func.19">`1</a> 
 ;<br/><a NAME="E2:73_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m1</font></span>)</span> <a href="coh_sp.html#K19" title="COH_SP:func.19">`1</a> 
 <b>by </b><i><a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th44</a></i>;<br/><a NAME="E3:73_1_1_1"/><b>then </b><i><font color="Green" title="E44">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="coh_sp.html#K19" title="COH_SP:func.19">`1</a> 
 <b>by </b><i><a class="txt" href="coh_sp.html#E1:73"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th44</a></i>;<br/><a NAME="E4:73_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m1</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m1</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  )
 <b>by </b><i><a class="txt" href="coh_sp.html#E1:73_1_1_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:73_1_1_1"/><b>then </b><i><font color="Green" title="E45">A6</font></i>: 
( <font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp; <font color="Maroon" title="c5">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="coh_sp.html#E1:73_1_1"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:73_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="coh_sp.html#K19" title="COH_SP:func.19">`1</a> 
 <b>by </b><i><a class="txt" href="coh_sp.html#E3:73_1_1_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E7:73_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c3">m2</font></span>)</span> <a href="coh_sp.html#K19" title="COH_SP:func.19">`1</a> 
 <b>by </b><i><a class="txt" href="coh_sp.html#E5:73_1_1_1"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:73_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c3">m2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K21" title="COH_SP:func.21">MapsT</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="coh_sp.html#E1:73"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="coh_sp.html#E1:73_1_1"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="coh_sp.html#E2:73_1_1"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="coh_sp.html#T42" target="_self" title="COH_SP:th.42">Th42</a></i>;<br/>


</div>
