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<b>consider </b><font color="Maroon" title="c3">f</font> being  <a href="cohsp_1.html#V10" title="COHSP_1:attr.10">U-linear</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/>
<b>set </b><font color="Maroon" title="c4">C</font> =  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>;<br/>
<a NAME="E2:117_1_1"/>
 <a href="cohsp_1.html#K11" title="COHSP_1:func.11">LinTrace</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>
 
<b>by </b><i><a class="ref" href="cohsp_1.html#D21" target="_self" title="COHSP_1:def.21">Def21</a></i>;<br/>
<b>hence </b><a NAME="E3:117_1_1"/>
not  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> is <a href="classes1.html#V1" title="CLASSES1:attr.1">subset-closed</a> &amp;  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> is <a href="coh_sp.html#V1" title="COH_SP:attr.1">binary_complete</a> )</span><br/>

<b>thus </b><a NAME="E4:117_1_1"/>
 <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> is <a href="classes1.html#V1" title="CLASSES1:attr.1">subset-closed</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> is <a href="coh_sp.html#V1" title="COH_SP:attr.1">binary_complete</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> or  not <font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font> or <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/>

<b>assume </b><a NAME="E1:117_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font> or <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c7">f</font> being  <a href="cohsp_1.html#V10" title="COHSP_1:attr.10">U-linear</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><b> such that </b><br/><a NAME="E3:117_1_1_1"/><i><font color="Green" title="E66">A1</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cohsp_1.html#K11" title="COHSP_1:func.11">LinTrace</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="ref" href="cohsp_1.html#D21" target="_self" title="COHSP_1:def.21">Def21</a></i>;<br/>
<b>assume </b><a NAME="E4:117_1_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font></span><br/>

<a NAME="E5:117_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">g</font> being <a href="cohsp_1.html#V10" title="COHSP_1:attr.10">U-linear</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> st  <a href="cohsp_1.html#K11" title="COHSP_1:func.11">LinTrace</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="#E3:117_1_1_1"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="cohsp_1.html#T62" target="_self" title="COHSP_1:th.62">Th62</a></i>;<br/>
<b>hence </b><a NAME="E6:117_1_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>
 <b>by </b><i><a class="ref" href="cohsp_1.html#D21" target="_self" title="COHSP_1:def.21">Def21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="cohsp_1.html#D1" target="_self" title="COHSP_1:def.1">COHSP_1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> holds <br/><font color="Olive" title="b1">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ) implies  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/>

<b>assume </b><a NAME="E5:117_1_1"/><i><font color="Green" title="E66">A2</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> holds <br/><font color="Olive" title="b1">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> implies  ex <font color="Olive" title="b1">f</font> being <a href="cohsp_1.html#V10" title="COHSP_1:attr.10">U-linear</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> st <font color="Maroon" title="c6">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cohsp_1.html#K11" title="COHSP_1:func.11">LinTrace</a> <font color="Olive" title="b1">f</font> )</span><br/><b>assume </b><a NAME="E1:117_1_1_2"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being <a href="cohsp_1.html#V10" title="COHSP_1:attr.10">U-linear</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> st <font color="Maroon" title="c6">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cohsp_1.html#K11" title="COHSP_1:func.11">LinTrace</a> <font color="Olive" title="b1">f</font></span><br/><a NAME="E2:117_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>
 <b>by </b><i><a class="txt" href="#E5:117_1_1"><i><font color="Green" title="E66">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:117_1_1_2"/>
 ex <font color="Olive" title="b1">f</font> being <a href="cohsp_1.html#V10" title="COHSP_1:attr.10">U-linear</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> st <font color="Maroon" title="c6">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cohsp_1.html#K11" title="COHSP_1:func.11">LinTrace</a> <font color="Olive" title="b1">f</font>
 <b>by </b><i><a class="ref" href="cohsp_1.html#D21" target="_self" title="COHSP_1:def.21">Def21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E7:117_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">f</font> being <a href="cohsp_1.html#V10" title="COHSP_1:attr.10">U-linear</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> st  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cohsp_1.html#K11" title="COHSP_1:func.11">LinTrace</a> <font color="Olive" title="b1">f</font>
 
<b>by </b><i><a class="ref" href="cohsp_1.html#T63" target="_self" title="COHSP_1:th.63">Th63</a></i>;<br/>
<b>hence </b><a NAME="E8:117_1_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K12" title="COHSP_1:func.12">LinCoh</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>
 <b>by </b><i><a class="ref" href="cohsp_1.html#D21" target="_self" title="COHSP_1:def.21">Def21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
