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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>assume </b><a NAME="E1:50"/><i><font color="Green">E36</font></i>: 
<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>
 ;<br/>

<a NAME="E2:50"/><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>by </b><i>, </i>;<br/>
<a NAME="E3:50"/><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>by </b><i>, </i>;<br/>
<a NAME="E4:50"/><i><font color="Green">E65</font></i>: 
(  <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> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</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="circtrm1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E71</font></i>: <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font></span>)</span> = 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T160">RELAT_1:160</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font>
<b>by </b><i><a class="txt" href="circtrm1.html#E1:50"><i><font color="Green">E36</font></i></a>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>
;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T123">FUNCT_1:123</a></i>
<br/>.= 
 <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font>
<b>by </b><i>, , </i>
;<br/>

<b>thus </b><a NAME="E7:50"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="msafree2.html#K3">InnerVertices</a> <font color="Maroon">S1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K3">InnerVertices</a> <font color="Maroon">S2</font>
 <b>by </b><i/>;<br/>


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