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

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>

<b>given </b><font color="Maroon">G1</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font><b> such that </b><a NAME="E1:29_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:29_1_2"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">e</font>
 <b>and </b><br/><a NAME="E3:29_1_2"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 <b>and </b><br/><a NAME="E4:29_1_2"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  st <font color="Olive">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Olive">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="setwiseo.html#K6" target="_self">\</a> <font color="Olive">B'</font> holds <br/><font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>
 ;<br/>

<b>given </b><font color="Maroon">G2</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font><b> such that </b><a NAME="E5:29_1_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E6:29_1_2"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">e</font>
 <b>and </b><br/><a NAME="E7:29_1_2"/><i><font color="Green">E112</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 <b>and </b><br/><a NAME="E8:29_1_2"/><i><font color="Green">E113</font></i>: 
for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>  st <font color="Olive">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Olive">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="setwiseo.html#K6" target="_self">\</a> <font color="Olive">B'</font> holds <br/><font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_2_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_1"><b>suppose </b></a><a NAME="E1:29_1_2_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:29_1_2_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
 <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T6" target="_self">Th6</a>, <a class="txt" href="setwiseo.html#E2:29"><i><font color="Green">E15</font></i></a></i>;<br/><b>hence </b><font color="Maroon">x</font> = 
 <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font>
<b>by </b><i>, , <a class="ref" href="setwiseo.html#T14" target="_self">Th14</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="setwiseo.html#T14" target="_self">Th14</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_2"><b>suppose </b></a><a NAME="E1:29_1_2_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E2:29_1_2_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font>]
 ;<br/><a NAME="E3:29_1_2_1_2"/><i><font color="Green">E118</font></i>: 
for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font><br/> for <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B'</font>] &amp; not <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">B'</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b</font></span><a href="tarski.html#K1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B'</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:29_1_2_1_2_1"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font> )
 ;<br/>

<b>assume </b><a NAME="E2:29_1_2_1_2_1"/><i><font color="Green">E120</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font>
 ;<br/>

<b>assume </b><a NAME="E3:29_1_2_1_2_1"/>
<font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E4:29_1_2_1_2_1"/><b>then </b><i><font color="Green">E121</font></i>: 
( <font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:29_1_2_1_2_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="setwiseo.html#K6" target="_self">\</a> <font color="Maroon">B'</font>
 
<b>by </b><i><a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>assume </b><a NAME="E6:29_1_2_1_2_1"/>
<font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_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="E1:29_1_2_1_2_1_1"/>
( <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_2_1_2_1_1_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">B'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T12" target="_self">Th12</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_2_1_2_1_1_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="setwiseo.html#T18" target="_self">Th18</a>, <a class="ref" href="setwiseo.html#T19" target="_self">Th19</a></i>
<br/>.= 
<font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#T13" target="_self">Th13</a>, <a class="ref" href="setwiseo.html#T16" target="_self">Th16</a>, <a class="ref" href="setwiseo.html#T18" target="_self">Th18</a>, <a class="ref" href="setwiseo.html#T19" target="_self">Th19</a>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:29_1_2_1_2"/>
for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B'</font>]
 <b>from </b><i><a class="ref" href="setwiseo.html#S2" target="_self">SETWISEO:sch 2</a>(, );<br/></i><b>hence </b><a NAME="E5:29_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="setwiseo.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
