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

<b>let </b><font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">E</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g1</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">f2</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g2</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:55"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">f1</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">f2</font>
 <b>and </b><br/><a NAME="E2:55"/><i><font color="Green">E26</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> )
 <b>and </b><br/><a NAME="E3:55"/><i><font color="Green">E27</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font> )
 <b>and </b><br/><a NAME="E4:55"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">f1</font> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">E</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font>
 <b>and </b><br/><a NAME="E5:55"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">p</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> holds <br/><font color="Maroon">f1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font></span>)</span>
 <b>and </b><br/><a NAME="E6:55"/><i><font color="Green">E31</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> )
 <b>and </b><br/><a NAME="E7:55"/><i><font color="Green">E32</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font> )
 <b>and </b><br/><a NAME="E8:55"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">E</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g2</font>
 <b>and </b><br/><a NAME="E9:55"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">p</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> holds <br/><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a><br/>

<b>set </b><font color="Maroon">f</font> = <font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">g1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g2</font>;<br/>
<b>set </b><font color="Maroon">S</font> = <font color="Maroon">S1</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">S2</font>;<br/>
<a NAME="E10:55"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">S2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#D2">CIRCCOMB:def 2</a></i>;<br/>
<b>hence </b><a NAME="E11:55"/>
 <a href="relat_1.html#K1">dom</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> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">S2</font></span>)</span>
 <b>by </b><i><a class="ref" href="algspec1.html#T6" target="_self">Th6</a>, <a class="ref" href="algspec1.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a><br/>

<a NAME="E12:55"/><i><font color="Green">E90</font></i>: 
the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">S2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#D2">CIRCCOMB:def 2</a></i>;<br/>
<b>hence </b><a NAME="E13:55"/>
 <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> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">S2</font></span>)</span>
 <b>by </b><i><a class="ref" href="algspec1.html#T6" target="_self">Th6</a>, <a class="ref" href="algspec1.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>

<a NAME="E14:55"/><i><font color="Green">E91</font></i>: 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_4.html#K1">+*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">S2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="circcomb.html#D2">CIRCCOMB:def 2</a></i>;<br/>
<a NAME="E15:55"/>
(  <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> <a href="tarski.html#R1">c=</a> <span class="p1">(<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="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font></span>)</span> &amp; <span class="p1">(<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="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font> )
 
<b>by </b><i>, <a class="ref" href="algspec1.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_4.html#T18">FUNCT_4:18</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E16:55"/>
 <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> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<a NAME="E17:55"/>
(  <a href="relat_1.html#K2">rng</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> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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#K2">rng</a> <font color="Maroon">g2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font> )
 
<b>by </b><i>, <a class="ref" href="algspec1.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_4.html#T18">FUNCT_4:18</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E18:55"/>
 <a href="relat_1.html#K2">rng</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> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<a NAME="E19:55"/><i><font color="Green">E92</font></i>: 
(  <a href="relset_1.html#K5">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> )
 
;<br/>
<a NAME="E20:55"/><i><font color="Green">E93</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </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> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">S2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</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> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="algspec1.html#T6" target="_self">Th6</a>, <a class="ref" href="algspec1.html#T7" target="_self">Th7</a>, <a class="ref" href="algspec1.html#T12" target="_self">Th12</a>, <a class="ref" href="algspec1.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_4.html#T73">FUNCT_4:73</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">E</font> <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>
<b>by </b><i>, , <a class="ref" href="algspec1.html#T8" target="_self">Th8</a>, <a class="ref" href="algspec1.html#T9" target="_self">Th9</a>, <a class="ref" href="algspec1.html#T15" target="_self">Th15</a>, <a class="ref" href="funct_7.html#T10">FUNCT_7:10</a></i>
;<br/>

<b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E22:55"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">S2</font></span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">S2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> )
 ;<br/>

<a NAME="E23:55"/><i><font color="Green">E98</font></i>: 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S1</font> <a href="funct_4.html#K1">+*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">S2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="circcomb.html#D2">CIRCCOMB:def 2</a></i>;<br/>
<a NAME="E24:55"/><i><font color="Green">E101</font></i>: 
(  <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> <a href="finseq_2.html#K3">*</a>  &amp;  <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> <a href="finseq_2.html#K3">*</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_2"/>
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> or  not <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1"><b>suppose </b></a><a NAME="E1:55_2_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>
 ;<br/><div class="add"><a NAME="E2:55_2_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="algspec1.html#T16" target="_self">Th16</a>, <a class="ref" href="algspec1.html#T18" target="_self">Th18</a>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><a NAME="E3:55_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font>
 <b>by </b><i>, <a class="ref" href="algspec1.html#T19" target="_self">Th19</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:55_2_1"/><b>then </b>
<font color="Maroon">p</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E5:55_2_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</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><a class="ref" href="algspec1.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E6:55_2_1"/><b>then </b><i><font color="Green">E124</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T44">RELAT_1:44</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><b>thus </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> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</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">p</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T11">FUNCT_7:11</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="algspec1.html#T20" target="_self">Th20</a>, <a class="ref" href="funct_4.html#T20">FUNCT_4:20</a></i>
<br/>.= 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="algspec1.html#T10" target="_self">Th10</a>, <a class="ref" href="algspec1.html#T19" target="_self">Th19</a>, </i>
<br/>.= 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="algspec1.html#T7" target="_self">Th7</a>, <a class="ref" href="algspec1.html#T19" target="_self">Th19</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_2"><b>suppose </b></a><a NAME="E1:55_2_2"/><i><font color="Green">E125</font></i>: 
not <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>
 ;<br/><div class="add"><a NAME="E2:55_2_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="algspec1.html#T11" target="_self">Th11</a>, <a class="ref" href="algspec1.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:55_2_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="algspec1.html#T16" target="_self">Th16</a>, <a class="ref" href="algspec1.html#T18" target="_self">Th18</a>, , <a class="ref" href="algspec1.html#T19" target="_self">Th19</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E4:55_2_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S1</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:55_2_2"/><b>then </b>
<font color="Maroon">p</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E6:55_2_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="algspec1.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E7:55_2_2"/><b>then </b><i><font color="Green">E128</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T44">RELAT_1:44</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><b>thus </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> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T35">FUNCT_4:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T11">FUNCT_7:11</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="algspec1.html#T21" target="_self">Th21</a>, <a class="ref" href="funct_4.html#T20">FUNCT_4:20</a></i>
<br/>.= 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="algspec1.html#T20" target="_self">Th20</a></i>
<br/>.= 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="algspec1.html#T7" target="_self">Th7</a>, <a class="ref" href="algspec1.html#T19" target="_self">Th19</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
