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<b>let </b><font color="Maroon">A1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">A2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:24_1_2"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> iff  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; <font color="Olive">f</font> is <a href="orders_3.html#V5" target="_self">monotone</a> ) )
 <b>and </b><br/><a NAME="E2:24_1_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> iff  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; <font color="Olive">f</font> is <a href="orders_3.html#V5" target="_self">monotone</a> ) )
 ;<br/>

<b>thus </b><a NAME="E3:24_1_2"/>
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:24_1_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:24_1_2_1_1"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> holds <br/><font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:24_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/>

<a NAME="E2:24_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; <font color="Olive">f</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:24_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<a NAME="E2:24_1_2_1"/>
<font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:24_1_2_1_2"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> holds <br/><font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:24_1_2_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 ;<br/>

<a NAME="E2:24_1_2_1_2_1"/><b>then </b>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; <font color="Olive">f</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:24_1_2_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:24_1_2_1"/>
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="orders_3.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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


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