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

<b>let </b><font color="Maroon" title="c4">it1</font>, <font color="Maroon" title="c5">it2</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">it1</font> is <a href="fsm_1.html#V4" title="FSM_1:attr.4">final</a> &amp; <font color="Maroon" title="c5">it2</font> is <a href="fsm_1.html#V4" title="FSM_1:attr.4">final</a> implies <font color="Maroon" title="c4">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">it2</font> )</span><br/>

<b>assume </b><a NAME="E1:70_1_2"/><i><font color="Green" title="E50">A3</font></i>: 
( <font color="Maroon" title="c4">it1</font> is <a href="fsm_1.html#V4" title="FSM_1:attr.4">final</a> &amp; <font color="Maroon" title="c5">it2</font> is <a href="fsm_1.html#V4" title="FSM_1:attr.4">final</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">it2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1_2_1"/>
<font color="Maroon" title="c4">it1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">it2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:70_1_2_1"/><b>then </b>
not for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it1</font> iff <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">it2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:70_1_2_1"/><i><font color="Green" title="E51">A4</font></i>: 
( ( <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it1</font> &amp; not <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">it2</font> ) or ( not <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it1</font> &amp; <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">it2</font> ) )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_2_1_1"/>
( ( <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it1</font> &amp; not <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">it2</font> ) or ( not <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it1</font> &amp; <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">it2</font> ) )
 </span><b>by </b><i><a class="txt" href="fsm_1.html#E4:70_1_2_1"><i><font color="Green" title="E51">A4</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_2_1_1_1"/><i><font color="Green" title="E52">A5</font></i>: 
( <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it1</font> &amp; not <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">it2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c7">X</font> = <font color="Maroon" title="c6">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">tfsm</font> ;<br/><b>consider </b><font color="Maroon" title="c8">qx</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">tfsm</font><b> such that </b><br/><a NAME="E4:70_1_2_1_1_1"/><i><font color="Green" title="E53">A6</font></i>: 
<font color="Maroon" title="c8">qx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2_1_1_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="fsm_1.html#T11" target="_self" title="FSM_1:th.11">Th11</a></i>;<br/><a NAME="E5:70_1_2_1_1_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:70_1_2_1_1_1"/><i><font color="Green" title="E54">A7</font></i>: 
( <font color="Maroon" title="c8">qx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c9">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">it2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">Z</font> = <font color="Maroon" title="c9">Z</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">tfsm</font> <b>by </b><i><a class="txt" href="fsm_1.html#E7:70_1_2_1_1_1"><i><font color="Green" title="E54">A7</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c11">XZ</font> = <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Z</font>;<br/><b>set </b><font color="Maroon" title="c12">ZX</font> = <font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X</font>;<br/><b>set </b><font color="Maroon" title="c13">Y'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Z</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X</font></span>)</span>;<br/><a NAME="E9:70_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Z</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X</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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Z</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:70_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a></i>;<br/>
<a NAME="E3:70_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Z</font> &amp; <font color="Maroon" title="c10">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E4:70_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2_1_1_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="fsm_1.html#E7:70_1_2_1_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c14">qy</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">tfsm</font><b> such that </b><br/><a NAME="E11:70_1_2_1_1_1"/><i><font color="Green" title="E55">A8</font></i>: 
<font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Z</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">X</font> = <font color="Maroon" title="c7">X</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">it1</font> <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2_1_1_1"><i><font color="Green" title="E52">A5</font></i></a></i>;<br/><div><i><font color="Green" title="E56">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1_2_1_1_1_2"/><i><font color="Green" title="E57">A10</font></i>: 
<font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:70_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E58">A11</font></i>: 
( <font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font> &amp; not <font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:70_1_2_1_1_1_2"/>
not <font color="Maroon" title="c8">qx</font>,<font color="Maroon" title="c14">qy</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1_2_1_1_1_2_1"/>
<font color="Maroon" title="c8">qx</font>,<font color="Maroon" title="c14">qy</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c16">Z'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">it2</font><b> such that </b><br/><a NAME="E3:70_1_2_1_1_1_2_1"/><i><font color="Green" title="E59">A12</font></i>: 
( <font color="Maroon" title="c8">qx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z'</font> &amp; <font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z'</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_2_1_1_1_2_1_1"/>
( <font color="Maroon" title="c10">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Z'</font> or <font color="Maroon" title="c10">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">Z'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Z'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:70_1_2_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2_1_1_1_2"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="fsm_1.html#E3:70_1_2_1_1_1_2_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:70_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">Z'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:70_1_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">Z</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c16">Z'</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E7:70_1_2_1_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/><a NAME="E3:70_1_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">Z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:70_1_2_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E7:70_1_2_1_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="fsm_1.html#E3:70_1_2_1_1_1_2_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:70_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="fsm_1.html#E4:70_1_2_1_1_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="fsm_1.html#E2:70_1_2_1_1_1_2"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1_2_1_1_1_3"/><i><font color="Green" title="E57">A13</font></i>: 
<font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c15">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:70_1_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E58">A14</font></i>: 
( <font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> &amp; not <font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:70_1_2_1_1_1_3"/>
not <font color="Maroon" title="c8">qx</font>,<font color="Maroon" title="c14">qy</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1_2_1_1_1_3_1"/>
<font color="Maroon" title="c8">qx</font>,<font color="Maroon" title="c14">qy</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c16">Z'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">it1</font><b> such that </b><br/><a NAME="E3:70_1_2_1_1_1_3_1"/><i><font color="Green" title="E59">A15</font></i>: 
( <font color="Maroon" title="c8">qx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z'</font> &amp; <font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z'</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_2_1_1_1_3_1_1"/>
( <font color="Maroon" title="c15">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Z'</font> or <font color="Maroon" title="c15">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">Z'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_2_1_1_1_3_1_1_1"/>
<font color="Maroon" title="c15">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Z'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:70_1_2_1_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2_1_1_1_3"><i><font color="Green" title="E57">A13</font></i></a>, <a class="txt" href="fsm_1.html#E3:70_1_2_1_1_1_3_1"><i><font color="Green" title="E59">A15</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:70_1_2_1_1_1_3_1_1_2"/>
<font color="Maroon" title="c15">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">Z'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:70_1_2_1_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c16">Z'</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/><a NAME="E3:70_1_2_1_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">Z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:70_1_2_1_1_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:70_1_2_1_1_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="fsm_1.html#E3:70_1_2_1_1_1_3_1"><i><font color="Green" title="E59">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:70_1_2_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="fsm_1.html#E7:70_1_2_1_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="fsm_1.html#E2:70_1_2_1_1_1_3"><i><font color="Green" title="E58">A14</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E15:70_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E11:70_1_2_1_1_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="fsm_1.html#E13:70_1_2_1_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:70_1_2_1_1_2"/><i><font color="Green" title="E52">A16</font></i>: 
( not <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it1</font> &amp; <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">it2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c7">X</font> = <font color="Maroon" title="c6">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">tfsm</font> ;<br/><b>consider </b><font color="Maroon" title="c8">qx</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">tfsm</font><b> such that </b><br/><a NAME="E4:70_1_2_1_1_2"/><i><font color="Green" title="E53">A17</font></i>: 
<font color="Maroon" title="c8">qx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2_1_1_2"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="fsm_1.html#T11" target="_self" title="FSM_1:th.11">Th11</a></i>;<br/><a NAME="E5:70_1_2_1_1_2"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:70_1_2_1_1_2"/><i><font color="Green" title="E54">A18</font></i>: 
( <font color="Maroon" title="c8">qx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c9">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it1</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">Z</font> = <font color="Maroon" title="c9">Z</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">tfsm</font> <b>by </b><i><a class="txt" href="fsm_1.html#E7:70_1_2_1_1_2"><i><font color="Green" title="E54">A18</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c11">XZ</font> = <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Z</font>;<br/><b>set </b><font color="Maroon" title="c12">ZX</font> = <font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X</font>;<br/><b>set </b><font color="Maroon" title="c13">Y'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Z</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X</font></span>)</span>;<br/><a NAME="E9:70_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Z</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X</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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Z</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:70_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a></i>;<br/>
<a NAME="E3:70_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Z</font> &amp; <font color="Maroon" title="c10">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E4:70_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2_1_1_2"><i><font color="Green" title="E52">A16</font></i></a>, <a class="txt" href="fsm_1.html#E7:70_1_2_1_1_2"><i><font color="Green" title="E54">A18</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c14">qy</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">tfsm</font><b> such that </b><br/><a NAME="E11:70_1_2_1_1_2"/><i><font color="Green" title="E55">A19</font></i>: 
<font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Z</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">X</font> = <font color="Maroon" title="c7">X</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">it2</font> <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2_1_1_2"><i><font color="Green" title="E52">A16</font></i></a></i>;<br/><div><i><font color="Green" title="E56">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1_2_1_1_2_2"/><i><font color="Green" title="E57">A21</font></i>: 
<font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c10">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:70_1_2_1_1_2_2"/><b>then </b><i><font color="Green" title="E58">A22</font></i>: 
( <font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font> &amp; not <font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:70_1_2_1_1_2_2"/>
not <font color="Maroon" title="c8">qx</font>,<font color="Maroon" title="c14">qy</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1_2_1_1_2_2_1"/>
<font color="Maroon" title="c8">qx</font>,<font color="Maroon" title="c14">qy</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c16">Z'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">it1</font><b> such that </b><br/><a NAME="E3:70_1_2_1_1_2_2_1"/><i><font color="Green" title="E59">A23</font></i>: 
( <font color="Maroon" title="c8">qx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z'</font> &amp; <font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z'</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_2_1_1_2_2_1_1"/>
( <font color="Maroon" title="c10">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Z'</font> or <font color="Maroon" title="c10">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">Z'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c10">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Z'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:70_1_2_1_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2_1_1_2_2"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="fsm_1.html#E3:70_1_2_1_1_2_2_1"><i><font color="Green" title="E59">A23</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:70_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c10">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">Z'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:70_1_2_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">Z</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c16">Z'</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E7:70_1_2_1_1_2"><i><font color="Green" title="E54">A18</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/><a NAME="E3:70_1_2_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">Z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:70_1_2_1_1_2_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E7:70_1_2_1_1_2"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="fsm_1.html#E3:70_1_2_1_1_2_2_1"><i><font color="Green" title="E59">A23</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:70_1_2_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="fsm_1.html#E4:70_1_2_1_1_2"><i><font color="Green" title="E53">A17</font></i></a>, <a class="txt" href="fsm_1.html#E2:70_1_2_1_1_2_2"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1_2_1_1_2_3"/><i><font color="Green" title="E57">A24</font></i>: 
<font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c15">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:70_1_2_1_1_2_3"/><b>then </b><i><font color="Green" title="E58">A25</font></i>: 
( <font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Z</font> &amp; not <font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:70_1_2_1_1_2_3"/>
not <font color="Maroon" title="c8">qx</font>,<font color="Maroon" title="c14">qy</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1_2_1_1_2_3_1"/>
<font color="Maroon" title="c8">qx</font>,<font color="Maroon" title="c14">qy</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c16">Z'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">it2</font><b> such that </b><br/><a NAME="E3:70_1_2_1_1_2_3_1"/><i><font color="Green" title="E59">A26</font></i>: 
( <font color="Maroon" title="c8">qx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z'</font> &amp; <font color="Maroon" title="c14">qy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Z'</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_2_1_1_2_3_1_1"/>
( <font color="Maroon" title="c15">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Z'</font> or <font color="Maroon" title="c15">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">Z'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon" title="c15">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Z'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:70_1_2_1_1_2_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2_1_1_2_3"><i><font color="Green" title="E57">A24</font></i></a>, <a class="txt" href="fsm_1.html#E3:70_1_2_1_1_2_3_1"><i><font color="Green" title="E59">A26</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:70_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon" title="c15">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">Z'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:70_1_2_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c16">Z'</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/><a NAME="E3:70_1_2_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">Z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:70_1_2_1_1_2_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E4:70_1_2_1_1_2"><i><font color="Green" title="E53">A17</font></i></a>, <a class="txt" href="fsm_1.html#E3:70_1_2_1_1_2_3_1"><i><font color="Green" title="E59">A26</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:70_1_2_1_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E1:70_1_2"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="fsm_1.html#E7:70_1_2_1_1_2"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="fsm_1.html#E2:70_1_2_1_1_2_3"><i><font color="Green" title="E58">A25</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E15:70_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E11:70_1_2_1_1_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="fsm_1.html#E13:70_1_2_1_1_2"><i><font color="Green" title="E56">A20</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:70_1_2"/>
<font color="Maroon" title="c4">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">it2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
