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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">U0</font>, <font color="Maroon" title="c3">U1</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c4">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c5">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">U1</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c2">U0</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c5">e</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">h1</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font></span>)</span>,<font color="Maroon" title="c3">U1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="equation.html#D5" target="_self" title="EQUATION:def.5">EQUATION:def 5</a><br/>

<b>assume </b><a NAME="E3:49"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c6">h1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">U1</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">F</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">U1</font><b> such that </b><br/><a NAME="E5:49"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c7">F</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">U1</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:49"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="msualg_3.html#D13" title="MSUALG_3:def.13">MSUALG_3:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">G</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c3">U1</font>,<font color="Maroon" title="c2">U0</font><b> such that </b><br/><a NAME="E7:49"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c8">G</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> 
 ;<br/>
<b>set </b><font color="Maroon" title="c9">F1</font> = <font color="Maroon" title="c8">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c6">h1</font>;<br/>
<a NAME="E8:49"/>
<font color="Maroon" title="c8">G</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c3">U1</font>,<font color="Maroon" title="c2">U0</font>
 
<b>by </b><i><a class="txt" href="equation.html#E5:49"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="equation.html#E7:49"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="msualg_3.html#T14" title="MSUALG_3:th.14">MSUALG_3:14</a></i>;<br/>
<a NAME="E9:49"/><b>then </b>
<font color="Maroon" title="c8">G</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">U1</font>,<font color="Maroon" title="c2">U0</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T13" title="MSUALG_3:th.13">MSUALG_3:13</a></i>;<br/>
<a NAME="E10:49"/><b>then </b><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c8">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c6">h1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">U0</font>
 
<b>by </b><i><a class="txt" href="equation.html#E3:49"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="msualg_3.html#T10" title="MSUALG_3:th.10">MSUALG_3:10</a></i>;<br/>
<a NAME="E11:49"/>
( <font color="Maroon" title="c7">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> &amp; <font color="Maroon" title="c7">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> )
 
<b>by </b><i><a class="txt" href="equation.html#E5:49"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="msualg_3.html#T13" title="MSUALG_3:th.13">MSUALG_3:13</a></i>;<br/>
<a NAME="E12:49"/><b>then </b><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c8">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<b>by </b><i><a class="txt" href="equation.html#E7:49"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="msualg_3.html#D5" title="MSUALG_3:def.5">MSUALG_3:def 5</a></i>;<br/>
<a NAME="E13:49"/>
<font color="Maroon" title="c7">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>
 
<b>by </b><i><a class="txt" href="equation.html#E5:49"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="msualg_3.html#T13" title="MSUALG_3:th.13">MSUALG_3:13</a></i>;<br/>
<a NAME="E14:49"/><b>then </b><i><font color="Green" title="E31">A8</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 3</a></i>;<br/>
<a NAME="E15:49"/>
<font color="Maroon" title="c7">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>
 
<b>by </b><i><a class="txt" href="equation.html#E5:49"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="msualg_3.html#T13" title="MSUALG_3:th.13">MSUALG_3:13</a></i>;<br/>
<a NAME="E16:49"/><b>then </b><i><font color="Green" title="E32">A9</font></i>: 
<font color="Maroon" title="c7">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T1" title="MSUALG_3:th.1">MSUALG_3:1</a></i>;<br/>
<a NAME="E17:49"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c6">h1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>;<br/>
<b>then </b><i><font color="Green" title="E33">A10</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c6">h1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E12:49"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="equation.html#E14:49"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="equation.html#E16:49"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></i>
<br/>.= 
<font color="Maroon" title="c6">h1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
;<br/>
<a NAME="E19:49"/><i><font color="Green" title="E34">A11</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c6">h1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E20:49"/><b>then </b><i><font color="Green" title="E35">A12</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c6">h1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="equation.html#T31" target="_self" title="EQUATION:th.31">Th31</a></i>;<br/>
<a NAME="E21:49"/><i><font color="Green" title="E36">A13</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c6">h1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="equation.html#E19:49"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="equation.html#T32" target="_self" title="EQUATION:th.32">Th32</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">h1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c6">h1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E18:49"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="equation.html#E20:49"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c6">h1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E2:49"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="equation.html#E10:49"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="equation.html#D5" target="_self" title="EQUATION:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">h1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E18:49"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="equation.html#E21:49"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>


</div>
