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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">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K20" title="COH_SP:func.20">MapsT</a> <font color="Maroon" title="c1">X</font> holds <br/> ( ( <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="coh_sp.html#K17" title="COH_SP:func.17">`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#K21" title="COH_SP:func.21">dom</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="coh_sp.html#K17" title="COH_SP:func.17">`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="Olive" title="b1">m</font> <a href="xfamily.html#K2" title="XFAMILY: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"><span class="p2">(<span class="default"><a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="coh_sp.html#K17" title="COH_SP:func.17">`2</a></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="coh_sp.html#K17" title="COH_SP:func.17">`2</a></span>)</span> &amp; (  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">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#K21" title="COH_SP:func.21">dom</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`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="Olive" title="b1">m</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">m</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">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#K22" title="COH_SP:func.22">cod</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`1</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K20" title="COH_SP:func.20">MapsT</a> <font color="Maroon" title="c1">X</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">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K17" title="COH_SP:func.17">`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#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K17" title="COH_SP:func.17">`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">m</font> <a href="xfamily.html#K2" title="XFAMILY: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"><span class="p2">(<span class="default"><a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K17" title="COH_SP:func.17">`2</a></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K17" title="COH_SP:func.17">`2</a></span>)</span> &amp; (  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">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#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`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">m</font> <a href="xfamily.html#K2" title="XFAMILY: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">m</font> <a href="xfamily.html#K2" title="XFAMILY: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#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`1</a>  ) )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K19" title="COH_SP:func.19">FuncsT</a> <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c4">T1</font>, <font color="Maroon" title="c5">T2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K16" title="COH_SP:func.16">TOL</a> <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E2:74"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c2">m</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">T1</font>,<font color="Maroon" title="c5">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c3">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:74"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
( ( <font color="Maroon" title="c5">T2</font> <a href="coh_sp.html#K17" title="COH_SP:func.17">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c4">T1</font> <a href="coh_sp.html#K17" title="COH_SP:func.17">`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">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">T1</font> <a href="coh_sp.html#K17" title="COH_SP:func.17">`2</a></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">T2</font> <a href="coh_sp.html#K17" title="COH_SP:func.17">`2</a></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E4:74"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">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> <font color="Maroon" title="c4">T1</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`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"><font color="Maroon" title="c3">f</font> <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"><font color="Maroon" title="c3">f</font> <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> <font color="Maroon" title="c5">T2</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T40" target="_self" onmouseover="rs('coh_sp/T40')" onmouseout="rh()">Th39</a></span>;<br/>
<a NAME="E5:74"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
<font color="Maroon" title="c5">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74"><span class="lab"><font color="Green" title="E38">A1</font></span></a></span>;<br/>
<a NAME="E6:74"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  &amp; <font color="Maroon" title="c4">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c2">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74"><span class="lab"><font color="Green" title="E38">A1</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E7:74"/>
( ( <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K17" title="COH_SP:func.17">`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#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K17" title="COH_SP:func.17">`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">m</font> <a href="xfamily.html#K2" title="XFAMILY: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"><span class="p2">(<span class="default"><a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K17" title="COH_SP:func.17">`2</a></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K17" title="COH_SP:func.17">`2</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E3:74"><span class="lab"><font color="Green" title="E39">A2</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">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">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#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`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">m</font> <a href="xfamily.html#K2" title="XFAMILY: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">m</font> <a href="xfamily.html#K2" title="XFAMILY: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#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`1</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`1</a>  implies <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">m</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`1</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E8:74"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`1</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"> <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">m</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`1</a> </span><br/>

<span class="kw">hence </span><a NAME="E9:74"/>
<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">m</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E6:74"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E5:74"><span class="lab"><font color="Green" title="E41">A4</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/>


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