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<b>let </b><font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">C1</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">C2</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S2</font>;<br/>







<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">C1</font>,<font color="Maroon">C2</font> <a href="circtrm1.html#R5" target="_self">are_similar_wrt</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 ;<br/><b>hence </b><a NAME="E2:63_1"/>
<font color="Maroon">S1</font>,<font color="Maroon">S2</font> <a href="circtrm1.html#R1" target="_self">are_equivalent_wrt</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 <b>by </b><i/>;<br/><a NAME="E3:63_1"/>
<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="circtrm1.html#R4" target="_self">form_embedding_of</a> <font color="Maroon">C1</font>,<font color="Maroon">C2</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:63_1"/>
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">C2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font> )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E2:63"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>and </b><br/><a NAME="E3:63"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font>
 <b>and </b><br/><a NAME="E4:63"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> ,<font color="Maroon">g</font> <a href="funct_1.html#K2">"</a>  <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S2</font>,<font color="Maroon">S1</font>
 <b>and </b><br/><a NAME="E5:63"/><i><font color="Green">E71</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">C2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="circtrm1.html#D9" target="_self">CIRCTRM1:def 9</a><br/>

<b>thus </b><a NAME="E6:63"/>
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">C2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="ref" href="circtrm1.html#T2" target="_self">Th2</a>, <a class="ref" href="circtrm1.html#T3" target="_self">Th3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="circtrm1.html#D12" target="_self">CIRCTRM1:def 12</a>,<a class="ref" href="circtrm1.html#D13" target="_self">CIRCTRM1:def 13</a><br/>

<b>thus </b><a NAME="E7:63"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K2">"</a>  is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T62">FUNCT_1:62</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="circtrm1.html#D12" target="_self">CIRCTRM1:def 12</a><br/>

<b>thus </b><a NAME="E8:63"/>
<font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> ,<font color="Maroon">g</font> <a href="funct_1.html#K2">"</a>  <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S2</font>,<font color="Maroon">S1</font>
 <b>by </b><i/>;<br/>

<a NAME="E9:63"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
 
<b>by </b><i>, <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>;<br/>
<a NAME="E10:63"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>then </b><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> = 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C2</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<b>hence </b>the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C2</font> = 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C2</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T78">RELAT_1:78</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C1</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="circtrm1.html#T3" target="_self">Th3</a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
;<br/><br/>

<a NAME="E13:63"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>
 
<b>by </b><i>, <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>;<br/>
<a NAME="E14:63"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>then </b><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span> = 
 <a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">C2</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<b>hence </b>the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">C2</font> = 
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">C2</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T78">RELAT_1:78</a></i>
<br/>.= 
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">C1</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="circtrm1.html#T3" target="_self">Th3</a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
;<br/><br/>


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