<?xml version="1.0"?>
<div class="add">

<span class="kw">consider </span><font color="Maroon" title="c3">a9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font>;<br/>
<a NAME="E2:149_2_1"/>
<font color="Maroon" title="c3">a9</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cohsp_1.html#T87" target="_self" title="COHSP_1:th.87">Th87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:149_2_1"/>
not <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font> is  <a href="classes1.html#V1" title="CLASSES1:attr.1">subset-closed</a>  &amp; <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font> is  <a href="coh_sp.html#V1" title="COH_SP:attr.1">binary_complete</a>  )</span><br/>

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</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="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font> is  <a href="coh_sp.html#V1" title="COH_SP:attr.1">binary_complete</a> </span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</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="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font> &amp; <font color="Maroon" title="c5">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">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:149_2_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <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"> ( <font color="Maroon" title="c5">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">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">aa</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c7">bb</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E3:149_2_1_1"/><span class="lab"><font color="Green" title="E87">A37</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">aa</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c7">bb</font>
 <span class="kw">and </span><br/><a NAME="E4:149_2_1_1"/><span class="lab"><font color="Green" title="E88">A38</font></span>: 
( <font color="Maroon" title="c6">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c7">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cohsp_1.html#T88" target="_self" title="COHSP_1:th.88">Th88</a></span>;<br/><span class="kw">assume </span><a NAME="E5:149_2_1_1"/>
<font color="Maroon" title="c5">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">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:149_2_1_1"/><span class="lab"><font color="Green" title="E89">A39</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x1</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c9">y1</font>
 <span class="kw">and </span><br/><a NAME="E8:149_2_1_1"/><span class="lab"><font color="Green" title="E90">A40</font></span>: 
( <font color="Maroon" title="c8">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">aa</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">bb</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:149_2_1_1"><span class="lab"><font color="Green" title="E87">A37</font></span></a>, <a class="ref" href="cohsp_1.html#T80" target="_self" title="COHSP_1:th.80">Th80</a></span>;<br/><a NAME="E9:149_2_1_1"/><span class="lab"><font color="Green" title="E91">A41</font></span>: 
( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:149_2_1_1"><span class="lab"><font color="Green" title="E90">A40</font></span></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></span>;<br/><a NAME="E10:149_2_1_1"/>
( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:149_2_1_1"><span class="lab"><font color="Green" title="E88">A38</font></span></a>, <a class="txt" href="#E8:149_2_1_1"><span class="lab"><font color="Green" title="E90">A40</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:149_2_1_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:149_2_1_1"><span class="lab"><font color="Green" title="E89">A39</font></span></a>, <a class="txt" href="#E9:149_2_1_1"><span class="lab"><font color="Green" title="E91">A41</font></span></a>, <a class="ref" href="cohsp_1.html#T87" target="_self" title="COHSP_1:th.87">Th87</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">let </span><font color="Maroon" title="c4">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c4">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="c4">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> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font> ) implies  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">A1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">A2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> ;<br/>
<span class="kw">assume </span><a NAME="E5:149_2_1"/><span class="lab"><font color="Green" title="E87">A42</font></span>: 
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="c4">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="c4">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> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <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">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="149_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  implies <font color="Maroon" title="c7">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:149_2_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  implies <font color="Maroon" title="c7">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">ax</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E3:149_2_1_2"/><span class="lab"><font color="Green" title="E88">A43</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ax</font>
 <span class="kw">and </span><br/><a NAME="E4:149_2_1_2"/><span class="lab"><font color="Green" title="E89">A44</font></span>: 
 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 <font color="Olive" title="b1">b</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c9">ax</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c10">bx</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><br/><a NAME="E6:149_2_1_2"/><span class="lab"><font color="Green" title="E90">A45</font></span>: 
<font color="Maroon" title="c10">bx</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c9">ax</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:149_2_1_2"><span class="lab"><font color="Green" title="E89">A44</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E7:149_2_1_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</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="c7">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">ay</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E9:149_2_1_2"/><span class="lab"><font color="Green" title="E91">A46</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">ay</font>
 <span class="kw">and </span><br/><a NAME="E10:149_2_1_2"/><span class="lab"><font color="Green" title="E92">A47</font></span>: 
 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 <font color="Olive" title="b1">b</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c11">ay</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c12">by1</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><br/><a NAME="E12:149_2_1_2"/><span class="lab"><font color="Green" title="E93">A48</font></span>: 
<font color="Maroon" title="c12">by1</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c11">ay</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:149_2_1_2"><span class="lab"><font color="Green" title="E92">A47</font></span></a></span>;<br/><a NAME="E13:149_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">bx</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c9">ax</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">by1</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c11">ay</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:149_2_1"><span class="lab"><font color="Green" title="E87">A42</font></span></a>, <a class="txt" href="#E6:149_2_1_2"><span class="lab"><font color="Green" title="E90">A45</font></span></a>, <a class="txt" href="#E12:149_2_1_2"><span class="lab"><font color="Green" title="E93">A48</font></span></a></span>;<br/><a NAME="E14:149_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">bx</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">by1</font></span>)</span> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ax</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">ay</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cohsp_1.html#T82" target="_self" title="COHSP_1:th.82">Th82</a></span>;<br/><a NAME="E15:149_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c7">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font> or <font color="Maroon" title="c7">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:149_2_1_2"><span class="lab"><font color="Green" title="E88">A43</font></span></a>, <a class="txt" href="#E9:149_2_1_2"><span class="lab"><font color="Green" title="E91">A46</font></span></a>, <a class="ref" href="cohsp_1.html#T87" target="_self" title="COHSP_1:th.87">Th87</a></span>;<br/><span class="kw">hence </span><a NAME="E16:149_2_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T1" title="COH_SP:th.1">COH_SP:1</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>
<a NAME="E7:149_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A49</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cohsp_1.html#D1" target="_self" title="COHSP_1:def.1">Def1</a></span>;<br/>
<a NAME="E8:149_2_1"/><span class="lab"><font color="Green" title="E89">A50</font></span>: 
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="149_2_1_3"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>,<a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">A</font> )</span><br/><span class="kw">assume </span><a NAME="E1:149_2_1_3_1"/><span class="lab"><font color="Green" title="E90">A51</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">A</font></span><br/><a NAME="E2:149_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A52</font></span>: 
<font color="Maroon" title="c7">x</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">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cohsp_1.html#T76" target="_self" title="COHSP_1:th.76">Th76</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="149_2_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:149_2_1_3_1_1"/>
( ( <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  ) or ( <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:149_2_1_3_1"><span class="lab"><font color="Green" title="E90">A51</font></span></a>, <a class="ref" href="cohsp_1.html#T76" target="_self" title="COHSP_1:th.76">Th76</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="149_2_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:149_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E92">A53</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">A</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:149_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E93">A54</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">and </span><br/><a NAME="E4:149_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E94">A55</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">ax</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><br/><a NAME="E6:149_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E95">A56</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ax</font>
 <span class="kw">and </span><br/><a NAME="E7:149_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E96">A57</font></span>: 
 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="c2">C2</font> st <font color="Maroon" title="c9">ax</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:149_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E94">A55</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">bx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E9:149_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E97">A58</font></span>: 
<font color="Maroon" title="c9">ax</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c10">bx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:149_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E96">A57</font></span></a></span>;<br/><a NAME="E10:149_2_1_3_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ax</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c10">bx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:149_2_1_3_1"><span class="lab"><font color="Green" title="E91">A52</font></span></a>, <a class="txt" href="#E1:149_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E92">A53</font></span></a>, <a class="txt" href="#E3:149_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E93">A54</font></span></a>, <a class="txt" href="#E6:149_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E95">A56</font></span></a>, <a class="ref" href="cohsp_1.html#T77" target="_self" title="COHSP_1:th.77">Th77</a></span>;<br/><span class="kw">hence </span><a NAME="E11:149_2_1_3_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:149_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E97">A58</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="149_2_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:149_2_1_3_1_1_2"/><span class="lab"><font color="Green" title="E92">A59</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">A</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:149_2_1_3_1_1_2"/><span class="lab"><font color="Green" title="E93">A60</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">and </span><br/><a NAME="E4:149_2_1_3_1_1_2"/><span class="lab"><font color="Green" title="E94">A61</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">bx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E6:149_2_1_3_1_1_2"/><span class="lab"><font color="Green" title="E95">A62</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">bx</font>
 <span class="kw">and </span><br/><a NAME="E7:149_2_1_3_1_1_2"/><span class="lab"><font color="Green" title="E96">A63</font></span>: 
 ex <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> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c9">bx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:149_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E94">A61</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">ax</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><br/><a NAME="E9:149_2_1_3_1_1_2"/><span class="lab"><font color="Green" title="E97">A64</font></span>: 
<font color="Maroon" title="c10">ax</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c9">bx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:149_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E96">A63</font></span></a></span>;<br/><a NAME="E10:149_2_1_3_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">ax</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c9">bx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:149_2_1_3_1"><span class="lab"><font color="Green" title="E91">A52</font></span></a>, <a class="txt" href="#E1:149_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E92">A59</font></span></a>, <a class="txt" href="#E3:149_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E93">A60</font></span></a>, <a class="txt" href="#E6:149_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E95">A62</font></span></a>, <a class="ref" href="cohsp_1.html#T78" target="_self" title="COHSP_1:th.78">Th78</a></span>;<br/><span class="kw">hence </span><a NAME="E11:149_2_1_3_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:149_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E97">A64</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">A</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:149_2_1_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:149_2_1_3"/><span class="lab"><font color="Green" title="E90">A65</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">and </span><br/><a NAME="E5:149_2_1_3"/><span class="lab"><font color="Green" title="E91">A66</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E6:149_2_1_3"/>
<font color="Maroon" title="c8">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:149_2_1"><span class="lab"><font color="Green" title="E87">A42</font></span></a>, <a class="txt" href="#E5:149_2_1_3"><span class="lab"><font color="Green" title="E91">A66</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">aa</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c10">bb</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E8:149_2_1_3"/><span class="lab"><font color="Green" title="E92">A67</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">aa</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c10">bb</font>
 <span class="kw">and </span><br/><a NAME="E9:149_2_1_3"/>
( <font color="Maroon" title="c9">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c10">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cohsp_1.html#T88" target="_self" title="COHSP_1:th.88">Th88</a></span>;<br/>
<a NAME="E10:149_2_1_3"/>
<font color="Maroon" title="c10">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:149_2_1_3"><span class="lab"><font color="Green" title="E91">A66</font></span></a>, <a class="txt" href="#E8:149_2_1_3"><span class="lab"><font color="Green" title="E92">A67</font></span></a></span>;<br/>
<a NAME="E11:149_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A68</font></span>: 
<font color="Maroon" title="c10">bb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E12:149_2_1_3"/>
<font color="Maroon" title="c9">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:149_2_1_3"><span class="lab"><font color="Green" title="E91">A66</font></span></a>, <a class="txt" href="#E8:149_2_1_3"><span class="lab"><font color="Green" title="E92">A67</font></span></a></span>;<br/>
<a NAME="E13:149_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">aa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E14:149_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">aa</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c10">bb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:149_2_1_3"><span class="lab"><font color="Green" title="E93">A68</font></span></a>, <a class="ref" href="cohsp_1.html#T79" target="_self" title="COHSP_1:th.79">Th79</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:149_2_1_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:149_2_1_3"><span class="lab"><font color="Green" title="E90">A65</font></span></a>, <a class="txt" href="#E8:149_2_1_3"><span class="lab"><font color="Green" title="E92">A67</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>
<div><span class="lab"><font color="Green" title="E90">A69</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="149_2_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:149_2_1_4"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">AA</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">consider </span><font color="Maroon" title="c8">aa</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">AA</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:149_2_1_4"/><span class="lab"><font color="Green" title="E91">A70</font></span>: 
<font color="Maroon" title="c8">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">and </span><br/><a NAME="E6:149_2_1_4"/><span class="lab"><font color="Green" title="E92">A71</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">ax</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><br/><a NAME="E8:149_2_1_4"/><span class="lab"><font color="Green" title="E93">A72</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">ax</font>
 <span class="kw">and </span><br/><a NAME="E9:149_2_1_4"/><span class="lab"><font color="Green" title="E94">A73</font></span>: 
 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="c2">C2</font> st <font color="Maroon" title="c10">ax</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:149_2_1_4"><span class="lab"><font color="Green" title="E92">A71</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">bx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E11:149_2_1_4"/><span class="lab"><font color="Green" title="E95">A74</font></span>: 
<font color="Maroon" title="c10">ax</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c11">bx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:149_2_1_4"><span class="lab"><font color="Green" title="E94">A73</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E12:149_2_1_4"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">AA</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">consider </span><font color="Maroon" title="c13">bb</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">AA</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:149_2_1_4"/><span class="lab"><font color="Green" title="E96">A75</font></span>: 
<font color="Maroon" title="c13">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">y</font>
 <span class="kw">and </span><br/><a NAME="E17:149_2_1_4"/><span class="lab"><font color="Green" title="E97">A76</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font> :  ex <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> st <font color="Olive" title="b2">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">by1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E19:149_2_1_4"/><span class="lab"><font color="Green" title="E98">A77</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">by1</font>
 <span class="kw">and </span><br/><a NAME="E20:149_2_1_4"/><span class="lab"><font color="Green" title="E99">A78</font></span>: 
 ex <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> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c15">by1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:149_2_1_4"><span class="lab"><font color="Green" title="E97">A76</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">ay</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><br/><a NAME="E22:149_2_1_4"/><span class="lab"><font color="Green" title="E100">A79</font></span>: 
<font color="Maroon" title="c16">ay</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c15">by1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:149_2_1_4"><span class="lab"><font color="Green" title="E99">A78</font></span></a></span>;<br/><a NAME="E23:149_2_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">ax</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c11">bx</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ay</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c15">by1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:149_2_1"><span class="lab"><font color="Green" title="E87">A42</font></span></a>, <a class="txt" href="#E11:149_2_1_4"><span class="lab"><font color="Green" title="E95">A74</font></span></a>, <a class="txt" href="#E22:149_2_1_4"><span class="lab"><font color="Green" title="E100">A79</font></span></a></span>;<br/><a NAME="E24:149_2_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">ax</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">ay</font></span>)</span> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">bx</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">by1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cohsp_1.html#T82" target="_self" title="COHSP_1:th.82">Th82</a></span>;<br/><span class="kw">hence </span><a NAME="E25:149_2_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:149_2_1_4"><span class="lab"><font color="Green" title="E91">A70</font></span></a>, <a class="txt" href="#E8:149_2_1_4"><span class="lab"><font color="Green" title="E93">A72</font></span></a>, <a class="txt" href="#E16:149_2_1_4"><span class="lab"><font color="Green" title="E96">A75</font></span></a>, <a class="txt" href="#E19:149_2_1_4"><span class="lab"><font color="Green" title="E98">A77</font></span></a>, <a class="ref" href="cohsp_1.html#T87" target="_self" title="COHSP_1:th.87">Th87</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="149_2_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  implies <font color="Maroon" title="c7">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> )</span><br/><span class="kw">assume </span><a NAME="E1:149_2_1_5"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  implies <font color="Maroon" title="c7">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">ax</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><br/><a NAME="E3:149_2_1_5"/><span class="lab"><font color="Green" title="E91">A80</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ax</font>
 <span class="kw">and </span><br/><a NAME="E4:149_2_1_5"/><span class="lab"><font color="Green" title="E92">A81</font></span>: 
 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="c2">C2</font> st <font color="Maroon" title="c9">ax</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c10">bx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E6:149_2_1_5"/><span class="lab"><font color="Green" title="E93">A82</font></span>: 
<font color="Maroon" title="c9">ax</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c10">bx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:149_2_1_5"><span class="lab"><font color="Green" title="E92">A81</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E7:149_2_1_5"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</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="c7">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">ay</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><br/><a NAME="E9:149_2_1_5"/><span class="lab"><font color="Green" title="E94">A83</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">ay</font>
 <span class="kw">and </span><br/><a NAME="E10:149_2_1_5"/><span class="lab"><font color="Green" title="E95">A84</font></span>: 
 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="c2">C2</font> st <font color="Maroon" title="c11">ay</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c12">by1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E12:149_2_1_5"/><span class="lab"><font color="Green" title="E96">A85</font></span>: 
<font color="Maroon" title="c11">ay</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c12">by1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:149_2_1_5"><span class="lab"><font color="Green" title="E95">A84</font></span></a></span>;<br/><a NAME="E13:149_2_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">ax</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c10">bx</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ay</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Maroon" title="c12">by1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:149_2_1"><span class="lab"><font color="Green" title="E87">A42</font></span></a>, <a class="txt" href="#E6:149_2_1_5"><span class="lab"><font color="Green" title="E93">A82</font></span></a>, <a class="txt" href="#E12:149_2_1_5"><span class="lab"><font color="Green" title="E96">A85</font></span></a></span>;<br/><a NAME="E14:149_2_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">ax</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">ay</font></span>)</span> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">bx</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">by1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cohsp_1.html#T82" target="_self" title="COHSP_1:th.82">Th82</a></span>;<br/><a NAME="E15:149_2_1_5"/><span class="kw">then </span>
( <font color="Maroon" title="c7">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> or <font color="Maroon" title="c7">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:149_2_1_5"><span class="lab"><font color="Green" title="E91">A80</font></span></a>, <a class="txt" href="#E9:149_2_1_5"><span class="lab"><font color="Green" title="E94">A83</font></span></a>, <a class="ref" href="cohsp_1.html#T87" target="_self" title="COHSP_1:th.87">Th87</a></span>;<br/><span class="kw">hence </span><a NAME="E16:149_2_1_5"/>
<font color="Maroon" title="c7">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T1" title="COH_SP:th.1">COH_SP:1</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>
<a NAME="E11:149_2_1"/><span class="kw">then </span>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> :  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="c2">C2</font> st <font color="Olive" title="b1">a</font> <a href="cohsp_1.html#K14" title="COHSP_1:func.14">U+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cohsp_1.html#D1" target="_self" title="COHSP_1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:149_2_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> <a href="cohsp_1.html#K16" title="COHSP_1:func.16">"\/"</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:149_2_1"><span class="lab"><font color="Green" title="E88">A49</font></span></a>, <a class="txt" href="#E9:149_2_1"><span class="lab"><font color="Green" title="E90">A69</font></span></a>, <a class="txt" href="#E8:149_2_1"><span class="lab"><font color="Green" title="E89">A50</font></span></a>, <a class="ref" href="cohsp_1.html#T87" target="_self" title="COHSP_1:th.87">Th87</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>
