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<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="msafree2.html#V2" title="MSAFREE2:attr.2">Circuit-like</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/> for <font color="Olive" title="b3">C1</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b4">C2</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="circtrm1.html#R4" title="CIRCTRM1:pred.4">form_embedding_of</a> <font color="Olive" title="b3">C1</font>,<font color="Olive" title="b4">C2</font> holds <br/>for <font color="Olive" title="b5">o1</font> being  <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b6">o2</font> being  <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b6">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">o1</font> holds <br/> <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b6">o2</font>,<font color="Olive" title="b4">C2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b5">o1</font>,<font color="Olive" title="b3">C1</font></span><br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C1</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b2">C2</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="circtrm1.html#R4" title="CIRCTRM1:pred.4">form_embedding_of</a> <font color="Olive" title="b1">C1</font>,<font color="Olive" title="b2">C2</font> holds <br/>for <font color="Olive" title="b3">o1</font> being  <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b4">o2</font> being  <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b4">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">o1</font> holds <br/> <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b4">o2</font>,<font color="Olive" title="b2">C2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b3">o1</font>,<font color="Olive" title="b1">C1</font></span><br/><b>let </b><font color="Maroon" title="c5">C1</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C2</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="circtrm1.html#R4" title="CIRCTRM1:pred.4">form_embedding_of</a> <font color="Maroon" title="c5">C1</font>,<font color="Olive" title="b1">C2</font> holds <br/>for <font color="Olive" title="b2">o1</font> being  <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b3">o2</font> being  <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b3">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">o1</font> holds <br/> <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b3">o2</font>,<font color="Olive" title="b1">C2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b2">o1</font>,<font color="Maroon" title="c5">C1</font></span><br/><b>let </b><font color="Maroon" title="c6">C2</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="circtrm1.html#R4" title="CIRCTRM1:pred.4">form_embedding_of</a> <font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c6">C2</font> implies for <font color="Olive" title="b1">o1</font> being  <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b2">o2</font> being  <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b2">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o1</font> holds <br/> <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b2">o2</font>,<font color="Maroon" title="c6">C2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o1</font>,<font color="Maroon" title="c5">C1</font> )</span><br/>











<b>assume </b><b>that </b><br/><a NAME="E1:67"/>
( <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c4">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>and </b><br/><a NAME="E2:67"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>
 <b>and </b><br/><a NAME="E3:67"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c5">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="c6">C2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E4:67"/><i><font color="Green" title="E45">A2</font></i>: 
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c5">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="c6">C2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">g</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o1</font> being  <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b2">o2</font> being  <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b2">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o1</font> holds <br/> <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b2">o2</font>,<font color="Maroon" title="c6">C2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o1</font>,<font color="Maroon" title="c5">C1</font></span><br/>

<b>let </b><font color="Maroon" title="c7">o1</font> be   <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o2</font> being  <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">o1</font> holds <br/> <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o2</font>,<font color="Maroon" title="c6">C2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c5">C1</font></span><br/><b>let </b><font color="Maroon" title="c8">o2</font> be   <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <font color="Maroon" title="c2">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">o1</font> implies  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c6">C2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c5">C1</font> )</span><br/>



<b>assume </b><a NAME="E5:67"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c8">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">o1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c6">C2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c5">C1</font></span><br/>

<a NAME="E6:67"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">G1</font>
 
<b>by </b><i><a class="txt" href="circtrm1.html#E2:67"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E7:67"/>
 <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c6">C2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c5">C1</font>
 <b>by </b><i><a class="txt" href="circtrm1.html#E4:67"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="circtrm1.html#E5:67"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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