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<span class="kw">let </span><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font> be   <a href="coh_sp.html#NM1" title="COH_SP:NM.1">Coherence_Space</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="cohsp_1.html#V9" title="COHSP_1:attr.9">U-stable</a>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font><br/> for <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font><br/> for <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> iff  ex <font color="Olive" title="b4">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">b</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>  <a href="cohsp_1.html#K8" title="COHSP_1:func.8">Trace</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b4">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">a</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="cohsp_1.html#V9" title="COHSP_1:attr.9">U-stable</a>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> iff  ex <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">b</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>  <a href="cohsp_1.html#K8" title="COHSP_1:func.8">Trace</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">a</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> iff  ex <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">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="cohsp_1.html#K8" title="COHSP_1:func.8">Trace</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">y</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> iff  ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">b</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="cohsp_1.html#K8" title="COHSP_1:func.8">Trace</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b1">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font> ) )</span><br/>







<a NAME="E1:88"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">b</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="cohsp_1.html#K8" title="COHSP_1:func.8">Trace</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b1">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font> ) implies <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:88_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">b</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="cohsp_1.html#K8" title="COHSP_1:func.8">Trace</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b1">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:88_1"/>
<font color="Maroon" title="c6">b</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">and </span><br/><a NAME="E4:88_1"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">and </span><br/><a NAME="E5:88_1"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">and </span><br/><a NAME="E6:88_1"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="cohsp_1.html#T23" target="_self" title="COHSP_1:th.23">Th23</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c6">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c7">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c8">b</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="cohsp_1.html#K8" title="COHSP_1:func.8">Trace</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c8">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">c</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">c</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="c3">f</font> &amp; <font color="Maroon" title="c9">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">c</font> implies <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:88_1_1"/>
<font color="Maroon" title="c9">c</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="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E2:88_1_1"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">and </span><br/><a NAME="E3:88_1_1"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font></span><br/><a NAME="E4:88_1_1"/>
<font color="Maroon" title="c9">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E2:88_1_1"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E5:88_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:88_1"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E3:88_1_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:88_1_1"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88_1_1"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:88_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">b</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="cohsp_1.html#K8" title="COHSP_1:func.8">Trace</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E5:88_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="cohsp_1.html#T32" target="_self" title="COHSP_1:th.32">Th32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font></span><br/><span class="kw">thus </span><a NAME="E10:88_1"/>
<font color="Maroon" title="c8">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c6">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font><span class="kw"> such that </span><a NAME="E4:88"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">b</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="cohsp_1.html#K8" title="COHSP_1:func.8">Trace</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E5:88"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font></span><br/>

<a NAME="E6:88"/><span class="lab"><font color="Green" title="E49">A9</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="cohsp_1.html#T32" target="_self" title="COHSP_1:th.32">Th32</a></span>;<br/>
<a NAME="E7:88"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E5:88"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="cohsp_1.html#D12" target="_self" title="COHSP_1:def.12">Def12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:88"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:88"><span class="lab"><font color="Green" title="E49">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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