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<span class="kw">let </span><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="struct_0.html#V11" title="STRUCT_0:attr.11">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#V4" title="MSUALG_1:attr.4">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#V4" title="MSUALG_1:attr.4">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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C1</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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#V4" title="MSUALG_1:attr.4">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/><span class="kw">let </span><font color="Maroon" title="c5">C1</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C2</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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/><span class="kw">let </span><font color="Maroon" title="c6">C2</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:65"/>
<font color="Maroon" title="c3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E2:65"/>
<font color="Maroon" title="c4">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E3:65"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:65"/>
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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#U3" title="MSUALG_1:sel.3">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>
 <span class="kw">and </span><br/><a NAME="E5:65"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
 the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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#U4" title="MSUALG_1:sel.4">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>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="circtrm1.html#D12" target="_self" onmouseover="rs('circtrm1/D12')" onmouseout="rh()">CIRCTRM1:def 12</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



<span class="kw">assume </span><a NAME="E6:65"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E7:65"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E40">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:65"/>
 <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:65"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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