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<b>let </b><font color="Maroon">a</font> be   <a href="triang_1.html#NM2">SetSequence</a>, <font color="Maroon">b</font> be   <a href="triang_1.html#NM2">SetSequence</a>;<br/>

<b>assume </b><a NAME="E1:30_1_2"/><i><font color="Green">E35</font></i>: 
( ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> ) &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:30_1_2_1"/>
( <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:30_1_2_1"/>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E3:30_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">a</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1_2_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:30_1_2_2"/>
( <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:30_1_2_2"/>
<font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E5:30_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:30_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D13">PBOOLE:def 13</a></i>;<br/>


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