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<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">A</font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/>



<b>assume </b><a NAME="E1:38"/><i><font color="Green">E39</font></i>: 
 ex <font color="Olive">X</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> st the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Olive">X</font></span><a href="pzfmisc1.html#K1">}</a></span>
 ;<br/>

<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">SB</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">ST</font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K8">Trivial_Algebra</a> <font color="Maroon">S</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:38"/><i><font color="Green">E42</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">X</font></span><a href="pzfmisc1.html#K1">}</a></span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">F</font> being    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K8">Trivial_Algebra</a> <font color="Maroon">S</font></span>)</span>;<br/>
<b>take </b>
<font color="Maroon">F</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D13">MSUALG_3:def 13</a><br/>

<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">X</font></span><a href="pzfmisc1.html#K1">}</a></span>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K8">Trivial_Algebra</a> <font color="Maroon">S</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E6:38"/>
<font color="Maroon">A</font> is <a href="msualg_1.html#V5">non-empty</a>
 
<b>by </b><i>, <a class="ref" href="msualg_1.html#D8">MSUALG_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:38"/>
<font color="Maroon">F</font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">A</font>, <a href="msafree2.html#K8">Trivial_Algebra</a> <font color="Maroon">S</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D12">MSUALG_3:def 12</a><br/>

<b>hence </b><a NAME="E8:38"/>
<font color="Maroon">F</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">A</font>, <a href="msafree2.html#K8">Trivial_Algebra</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D11">MSUALG_3:def 11</a><br/>

<a NAME="E9:38"/>
<font color="Maroon">F1</font> is <a href="msualg_3.html#V1">"1-1"</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:38"/>
<font color="Maroon">F</font> is <a href="msualg_3.html#V1">"1-1"</a>
 ;<br/>


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