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

<b>let </b><font color="Maroon">seq1</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>, <font color="Maroon">seq2</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>, <font color="Maroon">seq3</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<a NAME="E1:19"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">seq1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">seq2</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq3</font> implies for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">seq3</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon">seq1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">seq2</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq3</font>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">seq1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">seq2</font> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="comseq_1.html#K5">-</a> <font color="Maroon">seq3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D10">COMSEQ_1:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_1.html#K5">-</a> <font color="Maroon">seq3</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">seq3</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9">COMSEQ_1:def 9</a></i>
;<br/><b>hence </b><a NAME="E2:19_1_1"/>
<font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">seq3</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:19_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">seq3</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:19"/>
( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">seq3</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> ) implies <font color="Maroon">seq1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">seq2</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">seq3</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 ;<br/>

<a NAME="E2:19_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">seq1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:19_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">seq3</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">seq3</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_1.html#K5">-</a> <font color="Maroon">seq3</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D9">COMSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K2">+</a> <span class="p2">(<span class="default"><a href="comseq_1.html#K5">-</a> <font color="Maroon">seq3</font></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E4:19_2_1"/>
<font color="Maroon">seq1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="comseq_1.html#D10">COMSEQ_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:19_2"/>
<font color="Maroon">seq1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">seq2</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq3</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:19"/>
( <font color="Maroon">seq1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">seq2</font> <a href="comseq_1.html#K6">-</a> <font color="Maroon">seq3</font> iff for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">seq3</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div>
