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<span class="kw">let </span><font color="Maroon" title="c2">h1</font>, <font color="Maroon" title="c3">h2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font></span>)</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"> ( (  for <font color="Olive" title="b1">l2</font>, <font color="Olive" title="b2">l1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l2</font>,<font color="Olive" title="b2">l1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font> iff  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Olive" title="b1">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Olive" title="b2">l1</font> ) ) &amp; (  for <font color="Olive" title="b1">l2</font>, <font color="Olive" title="b2">l1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font>  st  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Olive" title="b1">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Olive" title="b2">l1</font> holds <br/><font color="Maroon" title="c2">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l2</font>,<font color="Olive" title="b2">l1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">l2</font> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <font color="Olive" title="b2">l1</font> ) &amp; (  for <font color="Olive" title="b1">l2</font>, <font color="Olive" title="b2">l1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l2</font>,<font color="Olive" title="b2">l1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font> iff  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Olive" title="b1">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Olive" title="b2">l1</font> ) ) &amp; (  for <font color="Olive" title="b1">l2</font>, <font color="Olive" title="b2">l1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font>  st  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Olive" title="b1">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Olive" title="b2">l1</font> holds <br/><font color="Maroon" title="c3">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l2</font>,<font color="Olive" title="b2">l1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">l2</font> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <font color="Olive" title="b2">l1</font> ) implies <font color="Maroon" title="c2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_3_2"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
 for <font color="Olive" title="b1">l2</font>, <font color="Olive" title="b2">l1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l2</font>,<font color="Olive" title="b2">l1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font> iff  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Olive" title="b1">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Olive" title="b2">l1</font> )
 <span class="kw">and </span><br/><a NAME="E2:42_3_2"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
 for <font color="Olive" title="b1">l2</font>, <font color="Olive" title="b2">l1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font>  st  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Olive" title="b1">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Olive" title="b2">l1</font> holds <br/><font color="Maroon" title="c2">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l2</font>,<font color="Olive" title="b2">l1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">l2</font> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <font color="Olive" title="b2">l1</font>
 <span class="kw">and </span><br/><a NAME="E3:42_3_2"/><span class="lab"><font color="Green" title="E24">A14</font></span>: 
 for <font color="Olive" title="b1">l2</font>, <font color="Olive" title="b2">l1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l2</font>,<font color="Olive" title="b2">l1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font> iff  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Olive" title="b1">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Olive" title="b2">l1</font> )
 <span class="kw">and </span><br/><a NAME="E4:42_3_2"/><span class="lab"><font color="Green" title="E25">A15</font></span>: 
 for <font color="Olive" title="b1">l2</font>, <font color="Olive" title="b2">l1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font>  st  <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Olive" title="b1">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Olive" title="b2">l1</font> holds <br/><font color="Maroon" title="c3">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l2</font>,<font color="Olive" title="b2">l1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">l2</font> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <font color="Olive" title="b2">l1</font>
 ; <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"> <font color="Maroon" title="c2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h2</font></span><br/>

<a NAME="E5:42_3_2"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_3_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="relat_1.html#D2" onmouseover="rs('relat_1/D2')" onmouseout="rh()">RELAT_1:def 2</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"> ( (  not <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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font> or <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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font> ) &amp; (  not <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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font> or <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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:42_3_2_1"/>
( <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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font> implies <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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font> )
  <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"> (  not <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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font> or <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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_3_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_3_2_1_1"/><span class="lab"><font color="Green" title="E27">A17</font></span>: 
<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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font>
 ; <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"> <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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">l2</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">l1</font> = <font color="Maroon" title="c5">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E3:42_3_2_1_1"/>
 <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c6">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c7">l1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_3_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E1:42_3_2_1_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:42_3_2_1_1"/>
<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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_3_2"><span class="lab"><font color="Green" title="E24">A14</font></span></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/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:42_3_2_1"/><span class="lab"><font color="Green" title="E27">A18</font></span>: 
<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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">h2</font>
 ; <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"> <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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">l2</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">l1</font> = <font color="Maroon" title="c5">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E4:42_3_2_1"/>
 <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c6">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c7">l1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_3_2"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="txt" href="#E2:42_3_2_1"><span class="lab"><font color="Green" title="E27">A18</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:42_3_2_1"/>
<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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_3_2"><span class="lab"><font color="Green" title="E22">A12</font></span></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/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_3_2_2"><span class="kw">now </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">  for <font color="Olive" title="b1">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <font color="Olive" title="b1">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font> holds <br/><font color="Maroon" title="c2">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">l</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</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"> ( <font color="Maroon" title="c4">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font> implies <font color="Maroon" title="c2">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">l</font> )</span><br/><span class="kw">assume </span><a NAME="E1:42_3_2_2"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
<font color="Maroon" title="c4">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">h1</font>
 ; <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"> <font color="Maroon" title="c2">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">l</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c5">l2</font>, <font color="Maroon" title="c6">l1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K5" title="COH_SP:func.5">MapsC</a> <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E3:42_3_2_2"/><span class="lab"><font color="Green" title="E28">A20</font></span>: 
<font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">l2</font>,<font color="Maroon" title="c6">l1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T1" onmouseover="rs('domain_1/T1')" onmouseout="rh()">DOMAIN_1:1</a></span>;<br/><a NAME="E4:42_3_2_2"/><span class="lab"><font color="Green" title="E29">A21</font></span>: 
 <a href="coh_sp.html#K6" title="COH_SP:func.6">dom</a> <font color="Maroon" title="c5">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K7" title="COH_SP:func.7">cod</a> <font color="Maroon" title="c6">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_3_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E1:42_3_2_2"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E3:42_3_2_2"><span class="lab"><font color="Green" title="E28">A20</font></span></a></span>;<br/><a NAME="E5:42_3_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">l2</font>,<font color="Maroon" title="c6">l1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l2</font> <a href="coh_sp.html#K9" title="COH_SP:func.9">*</a> <font color="Maroon" title="c6">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_3_2"><span class="lab"><font color="Green" title="E23">A13</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:42_3_2_2"/>
<font color="Maroon" title="c2">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_3_2"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E3:42_3_2_2"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E4:42_3_2_2"><span class="lab"><font color="Green" title="E29">A21</font></span></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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:42_3_2"/>
<font color="Maroon" title="c2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_3_2"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</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/>


</div>
