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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> , <font color="Maroon">S3</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">f1</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g1</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">f2</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g2</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/><b>let </b><font color="Maroon">C3</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S3</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:60"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g1</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f1</font>,<font color="Maroon">g1</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="E3:60"/><i><font color="Green">E42</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">f1</font>
 <b>and </b><br/><a NAME="E4:60"/><i><font color="Green">E65</font></i>: 
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">g1</font>
 <b>and </b><br/><a NAME="E5:60"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">f2</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g2</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>and </b><br/><a NAME="E6:60"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">f2</font>,<font color="Maroon">g2</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S2</font>,<font color="Maroon">S3</font>
 <b>and </b><br/><a NAME="E7:60"/><i><font color="Green">E73</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C3</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f2</font>
 <b>and </b><br/><a NAME="E8:60"/><i><font color="Green">E75</font></i>: 
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">C3</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g2</font>
 ; <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="E9:60"/>
( <font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i>, <a class="ref" href="circtrm1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T46">FUNCT_1:46</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="E10:60"/>
<font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font>,<font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S1</font>,<font color="Maroon">S3</font>
 <b>by </b><i>, , <a class="ref" href="pua2mss1.html#T30">PUA2MSS1:30</a></i>;<br/>

<b>thus </b><a NAME="E11:60"/>
( 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">C3</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span> &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">C3</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="circtrm1.html#E3:60"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="circtrm1.html#T3" target="_self">Th3</a>, <a class="ref" href="circtrm1.html#T4" target="_self">Th4</a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>


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