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

<b>let </b><font color="Maroon">f1</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">f2</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">f1</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">f2</font>
 ;<br/>

<a NAME="E2:29"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">f2</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#T31">FUNCT_4:31</a></i>;<br/>
<b>let </b><font color="Maroon">g1</font> be    <a href="algspec1.html#M1" target="_self">rng-retract</a> of <font color="Maroon">f1</font>;<br/><b>let </b><font color="Maroon">g2</font> be    <a href="algspec1.html#M1" target="_self">rng-retract</a> of <font color="Maroon">f2</font>;<br/>



<a NAME="E3:29"/><i><font color="Green">E27</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> )
 
<b>by </b><i><a class="ref" href="algspec1.html#T1" target="_self">Th1</a></i>;<br/>
<b>thus </b> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E2:29"><i><font color="Green">E26</font></i></a>, <a class="txt" href="algspec1.html#E3:29"><i><font color="Green">E27</font></i></a>, <a class="ref" href="relat_1.html#T26">RELAT_1:26</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="algspec1.html#D2" target="_self">ALGSPEC1:def 2</a><br/>

<a NAME="E5:29"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g2</font></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E1:29"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_4.html#T73">FUNCT_4:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g2</font></span>)</span>
<b>by </b><i><a class="ref" href="algspec1.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="algspec1.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T23">FUNCT_4:23</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E2:29"><i><font color="Green">E26</font></i></a>, <a class="ref" href="relat_1.html#T26">RELAT_1:26</a></i>
;<br/><br/>


</div>
