<?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> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" target="_self">Equations</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/><b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">I</font>,<font color="Maroon">S</font>;<br/>









<b>assume </b><a NAME="E1:53"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font> <a href="equation.html#R1" target="_self">|=</a> <font color="Maroon">e</font>
 ;<br/>

<b>let </b><font color="Maroon">h</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="equation.html#D5" target="_self">EQUATION:def 5</a><br/>

<b>assume </b><a NAME="E2:53"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">h</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font>, <a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:53"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">e</font> <a href="mcart_1.html#K1">`1</a>  as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">e2</font> = <font color="Maroon">e</font> <a href="mcart_1.html#K2">`2</a>  as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <b>by </b><i/>;<br/>
<a NAME="E6:53"/><i><font color="Green">E40</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><b>set </b><font color="Maroon">Z</font> = <span class="p1">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">h</font>;<br/><a NAME="E1:53_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font> <a href="equation.html#R1" target="_self">|=</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/><a NAME="E2:53_1"/>
 <a href="pralg_3.html#K4">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font>,<font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pralg_3.html#T25">PRALG_3:25</a></i>;<br/><a NAME="E3:53_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">h</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font>,<font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="msualg_3.html#T10">MSUALG_3:10</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e1</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#D3">PRALG_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e1</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">h</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e1</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">h</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e2</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e2</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e2</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#D3">PRALG_3:def 3</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:53"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="msualg_9.html#T15">MSUALG_9:15</a></i>;<br/>


</div>
