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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="instalg1.html#V1" title="INSTALG1:attr.1">feasible</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">S'</font> be  non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="instalg1.html#M1" title="INSTALG1:mode.1">Subsignature</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c4">A2</font> be    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/>





<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E25">A1</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A1</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">A2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">h</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c4">A2</font>;<br/>

<b>assume </b><a NAME="E2:45"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c4">A2</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">f</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S'</font>;<br/>
<b>set </b><font color="Maroon" title="c7">g</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S'</font>;<br/>
<a NAME="E3:45"/><i><font color="Green" title="E27">A3</font></i>: 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S'</font>, <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S'</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="instalg1.html#D2" target="_self" title="INSTALG1:def.2">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">f</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S'</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S'</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="instalg1.html#T10" target="_self" title="INSTALG1:th.10">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">h'</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">A1</font> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c8">f</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S'</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">A2</font> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c8">f</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S'</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:45"/><i><font color="Green" title="E28">A4</font></i>: 
( <font color="Maroon" title="c9">h'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">h</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c9">h'</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">A1</font> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c8">f</font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S'</font></span>)</span>,<font color="Maroon" title="c4">A2</font> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c8">f</font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="instalg1.html#E1:45"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="instalg1.html#E2:45"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="instalg1.html#E3:45"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="instalg1.html#T35" target="_self" title="INSTALG1:th.35">Th35</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">k</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">A1</font> <a href="instalg1.html#K2" title="INSTALG1:func.2">|</a> <font color="Maroon" title="c2">S'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">A2</font> <a href="instalg1.html#K2" title="INSTALG1:func.2">|</a> <font color="Maroon" title="c2">S'</font></span>)</span><b> such that </b><br/><a NAME="E8:45"/><i><font color="Green" title="E29">A5</font></i>: 
( <font color="Maroon" title="c10">k</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c9">h'</font> &amp; <font color="Maroon" title="c10">k</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">A1</font> <a href="instalg1.html#K2" title="INSTALG1:func.2">|</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c4">A2</font> <a href="instalg1.html#K2" title="INSTALG1:func.2">|</a> <font color="Maroon" title="c2">S'</font> )
 <b>by </b><i><a class="txt" href="instalg1.html#E6:45"><i><font color="Green" title="E28">A4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">k</font>
;<br/>

<b>thus </b><a NAME="E9:45"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S'</font> &amp; <font color="Maroon" title="c10">k</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">A1</font> <a href="instalg1.html#K2" title="INSTALG1:func.2">|</a> <font color="Maroon" title="c2">S'</font>,<font color="Maroon" title="c4">A2</font> <a href="instalg1.html#K2" title="INSTALG1:func.2">|</a> <font color="Maroon" title="c2">S'</font> )
 <b>by </b><i><a class="txt" href="instalg1.html#E6:45"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="instalg1.html#E8:45"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="relat_1.html#T94" title="RELAT_1:th.94">RELAT_1:94</a></i>;<br/>


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