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<span class="kw">reconsider </span><font color="Maroon" title="c2">A</font> =  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E2:25_1_1"/>
<font color="Maroon" title="c2">A</font> is   <a href="coh_sp.html#NM1" title="COH_SP:NM.1">Coherence_Space</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T2" target="_self" onmouseover="rs('coh_sp/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E3:25_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">x</font> is   <a href="coh_sp.html#NM1" title="COH_SP:NM.1">Coherence_Space</a> </span> } </span> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c2">A</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K3" title="COH_SP:func.3">CSp</a> <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">F</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">T</font></span>)</span>,<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b2">TT</font></span>)</span>)</span>)</span> where <font color="Olive" title="b1">T</font>, <font color="Olive" title="b2">TT</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K3" title="COH_SP:func.3">CSp</a> <font color="Maroon" title="c1">X</font> : verum </span> } </span> ;<br/>
<a NAME="E5:25_1_1"/>
(  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">A</font></span>)</span>) &amp;  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">A</font></span>)</span>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">T</font></span>)</span>,<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b2">TT</font></span>)</span>)</span>)</span> where <font color="Olive" title="b1">T</font>, <font color="Olive" title="b2">TT</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K3" title="COH_SP:func.3">CSp</a> <font color="Maroon" title="c1">X</font> : verum </span> } </span>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T9" onmouseover="rs('funct_2/T9')" onmouseout="rh()">FUNCT_2:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">UF</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">T</font></span>)</span>,<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b2">TT</font></span>)</span>)</span>)</span> where <font color="Olive" title="b1">T</font>, <font color="Olive" title="b2">TT</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K3" title="COH_SP:func.3">CSp</a> <font color="Maroon" title="c1">X</font> : verum </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><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">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">UF</font> holds <br/><font color="Olive" title="b1">f</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">UF</font> implies <font color="Maroon" title="c6">f</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )</span><br/><span class="kw">assume </span><a NAME="E1:25_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">UF</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="c6">f</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">C</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:25_1_1_1"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 <span class="kw">and </span><br/><a NAME="E4:25_1_1_1"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c7">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">T</font></span>)</span>,<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b2">TT</font></span>)</span>)</span>)</span> where <font color="Olive" title="b1">T</font>, <font color="Olive" title="b2">TT</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K3" title="COH_SP:func.3">CSp</a> <font color="Maroon" title="c1">X</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><a NAME="E5:25_1_1_1"/>
 ex <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K3" title="COH_SP:func.3">CSp</a> <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b2">B</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:25_1_1_1"/>
<font color="Maroon" title="c6">f</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_1_1_1"><span class="lab"><font color="Green" title="E11">A1</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="E8:25_1_1"/>
(  not  <a href="coh_sp.html#K4" title="COH_SP:func.4">FuncsC</a> <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="coh_sp.html#K4" title="COH_SP:func.4">FuncsC</a> <font color="Maroon" title="c1">X</font> is  <a href="funct_1.html#V4" title="FUNCT_1:attr.4">functional</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D13" onmouseover="rs('funct_1/D13')" onmouseout="rh()">FUNCT_1:def 13</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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