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<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font>, <font color="Maroon" title="c3">S3</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">g1</font>, <font color="Maroon" title="c6">f2</font>, <font color="Maroon" title="c7">g2</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c8">C1</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S1</font>;<br/><b>let </b><font color="Maroon" title="c9">C2</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S2</font>;<br/><b>let </b><font color="Maroon" title="c10">C3</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c3">S3</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:58"/><i><font color="Green" title="E38">A1</font></i>: 
( <font color="Maroon" title="c4">f1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c5">g1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>and </b><br/><a NAME="E2:58"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">g1</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font>
 <b>and </b><br/><a NAME="E3:58"/><i><font color="Green" title="E40">A3</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c8">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c9">C2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f1</font>
 <b>and </b><br/><a NAME="E4:58"/><i><font color="Green" title="E41">A4</font></i>: 
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c8">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c9">C2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">g1</font>
 <b>and </b><br/><a NAME="E5:58"/><i><font color="Green" title="E42">A5</font></i>: 
( <font color="Maroon" title="c6">f2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c7">g2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>and </b><br/><a NAME="E6:58"/><i><font color="Green" title="E43">A6</font></i>: 
<font color="Maroon" title="c6">f2</font>,<font color="Maroon" title="c7">g2</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c3">S3</font>
 <b>and </b><br/><a NAME="E7:58"/><i><font color="Green" title="E44">A7</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c9">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c10">C3</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">f2</font>
 <b>and </b><br/><a NAME="E8:58"/><i><font color="Green" title="E45">A8</font></i>: 
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c9">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c10">C3</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">g2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="circtrm1.html#D12" target="_self" title="CIRCTRM1:def.12">CIRCTRM1:def 12</a><br/>

<b>thus </b><a NAME="E9:58"/>
( <font color="Maroon" title="c6">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">g1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="circtrm1.html#E1:58"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="circtrm1.html#E5:58"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="funct_1.html#T46" title="FUNCT_1:th.46">FUNCT_1:46</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="circtrm1.html#D12" target="_self" title="CIRCTRM1:def.12">CIRCTRM1:def 12</a><br/>

<b>thus </b><a NAME="E10:58"/>
<font color="Maroon" title="c6">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">g1</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">S3</font>
 <b>by </b><i><a class="txt" href="circtrm1.html#E2:58"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="circtrm1.html#E6:58"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="pua2mss1.html#T30" title="PUA2MSS1:th.30">PUA2MSS1:30</a></i>;<br/>

<b>thus </b><a NAME="E11:58"/>
( the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c8">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c10">C3</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f1</font></span>)</span> &amp; the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c8">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c10">C3</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">g1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="circtrm1.html#E3:58"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="circtrm1.html#E4:58"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="circtrm1.html#E7:58"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="circtrm1.html#E8:58"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>;<br/>


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