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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#R6">are_isomorphic</a> 
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="equation.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>; <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="E3:49"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:49"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#R4">is_isomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E1:49"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msualg_3.html#D13">MSUALG_3:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:49"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#K4">""</a> 
 ;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E8:49"/>
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#R4">is_isomorphism</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E5:49"><i><font color="Green">E27</font></i></a>, <a class="txt" href="equation.html#E7:49"><i><font color="Green">E28</font></i></a>, <a class="ref" href="msualg_3.html#T14">MSUALG_3:14</a></i>;<br/>
<a NAME="E9:49"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T13">MSUALG_3:13</a></i>;<br/>
<a NAME="E10:49"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E3:49"><i><font color="Green">E26</font></i></a>, <a class="ref" href="msualg_3.html#T10">MSUALG_3:10</a></i>;<br/>
<a NAME="E11:49"/>
( <font color="Maroon">c<sub>7</sub></font> is <a href="msualg_3.html#V1">"1-1"</a> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="msualg_3.html#V2">"onto"</a> )
 
<b>by </b><i><a class="txt" href="equation.html#E5:49"><i><font color="Green">E27</font></i></a>, <a class="ref" href="msualg_3.html#T13">MSUALG_3:13</a></i>;<br/>
<a NAME="E12:49"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i><a class="txt" href="equation.html#E7:49"><i><font color="Green">E28</font></i></a>, <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>;<br/>
<a NAME="E13:49"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="msualg_3.html#V2">"onto"</a>
 
<b>by </b><i><a class="txt" href="equation.html#E5:49"><i><font color="Green">E27</font></i></a>, <a class="ref" href="msualg_3.html#T13">MSUALG_3:13</a></i>;<br/>
<a NAME="E14:49"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/>
<a NAME="E15:49"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="msualg_3.html#V1">"1-1"</a>
 
<b>by </b><i><a class="txt" href="equation.html#E5:49"><i><font color="Green">E27</font></i></a>, <a class="ref" href="msualg_3.html#T13">MSUALG_3:13</a></i>;<br/>
<a NAME="E16:49"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>;<br/>
<a NAME="E17:49"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>;<br/>
<b>then </b><i><font color="Green">E33</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K1">id</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="pboole.html#K7">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E12:49"><i><font color="Green">E30</font></i></a>, <a class="txt" href="equation.html#E14:49"><i><font color="Green">E31</font></i></a>, <a class="txt" href="equation.html#E16:49"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_2.html#T35">FUNCT_2:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
;<br/>
<a NAME="E19:49"/><i><font color="Green">E34</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E20:49"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="equation.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E21:49"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="equation.html#E19:49"><i><font color="Green">E34</font></i></a>, <a class="ref" href="equation.html#T32" target="_self">Th32</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E18:49"><i><font color="Green">E33</font></i></a>, <a class="txt" href="equation.html#E20:49"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E2:49"><i><font color="Green">E25</font></i></a>, <a class="txt" href="equation.html#E10:49"><i><font color="Green">E29</font></i></a>, <a class="ref" href="equation.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E18:49"><i><font color="Green">E33</font></i></a>, <a class="txt" href="equation.html#E21:49"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>


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