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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S'</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>







<b>assume </b><a NAME="E1:46"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S'</font>,<font color="Maroon">S</font>
 ;<br/>

<b>let </b><font color="Maroon">B</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S'</font>;<br/>

<b>assume </b><a NAME="E2:46"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S'</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font>
 ;<br/>

<a NAME="E3:46"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S'</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> )
 
<b>by </b><i>, <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S'</font>,the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>let </b><font color="Maroon">s1</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S'</font>, <font color="Maroon">s2</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S'</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>given </b><font color="Maroon">o</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S'</font><b> such that </b><a NAME="E5:46"/><i><font color="Green">E52</font></i>: 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 <b>and </b><br/><a NAME="E6:46"/><i><font color="Green">E53</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp;  ex <font color="Olive">a</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="msualg_6.html#K4">transl</a> <font color="Maroon">o</font>,<font color="Olive">i</font>,<font color="Olive">a</font>,<font color="Maroon">B</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_6.html#D5">MSUALG_6:def 5</a><br/>

<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">a</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:46"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="msualg_6.html#K4">transl</a> <font color="Maroon">o</font>,<font color="Maroon">i</font>,<font color="Maroon">a</font>,<font color="Maroon">B</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_6.html#D5">MSUALG_6:def 5</a><br/>

<a NAME="E9:46"/>
( <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S'</font> <a href="funct_2.html#R2">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>;<br/>
<b>hence </b> <a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S'</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s2</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a>, </i>
;<br/><br/>

<b>take </b>
<font color="Maroon">i</font>
;<br/>

<a NAME="E11:46"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>;<br/>
<a NAME="E12:46"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S'</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S'</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E13:46"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>

<b>hence </b><i><font color="Green">E81</font></i>: <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><br/>

<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E15:46"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">A</font>
 <b>by </b><i>, , , </i>;<br/>

<a NAME="E16:46"/><i><font color="Green">E102</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K4">transl</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">a</font>,<font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> )
 
<b>by </b><i>, <a class="ref" href="instalg1.html#T2" target="_self">Th2</a>, <a class="ref" href="msualg_6.html#D4">MSUALG_6:def 4</a></i>;<br/>
<a NAME="E17:46"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E18:46"/><b>then </b><i><font color="Green">E105</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3"><b>now </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="E1:46_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 ;<br/><a NAME="E2:46_3"/><b>then </b>
( <font color="Maroon">t</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"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_6.html#K4">transl</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">a</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">x</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="instalg1.html#T2" target="_self">Th2</a>, , <a class="ref" href="msualg_6.html#D4">MSUALG_6:def 4</a></i>;<br/><b>hence </b><a NAME="E3:46_3"/>
<font color="Maroon">t</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"><a href="msualg_6.html#K4">transl</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">a</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:46"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="msualg_6.html#K4">transl</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">a</font>,<font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
