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<span class="kw">let </span><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A1</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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/>  for <font color="Olive" title="b2">A2</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S2</font>  st <font color="Olive" title="b1">A1</font> is  <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a>  &amp; <font color="Olive" title="b2">A2</font> is  <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a>  &amp;  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">A1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b2">A2</font> holds <br/><font color="Olive" title="b1">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Olive" title="b2">A2</font> is  <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">A1</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A2</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S2</font>  st <font color="Maroon" title="c3">A1</font> is  <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a>  &amp; <font color="Olive" title="b1">A2</font> is  <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a>  &amp;  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">A2</font> holds <br/><font color="Maroon" title="c3">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Olive" title="b1">A2</font> is  <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">A2</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A1</font> is  <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a>  &amp; <font color="Maroon" title="c4">A2</font> is  <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a>  &amp;  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">A2</font> implies <font color="Maroon" title="c3">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon" title="c4">A2</font> is  <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a>  )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c3">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon" title="c4">A2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">S</font> = <font color="Maroon" title="c1">S1</font> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <font color="Maroon" title="c2">S2</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:102"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
<font color="Maroon" title="c3">A1</font> is  <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> 
 <span class="kw">and </span><br/><a NAME="E2:102"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
<font color="Maroon" title="c4">A2</font> is  <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> 
 <span class="kw">and </span><br/><a NAME="E3:102"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">A2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon" title="c4">A2</font> is  <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> </span><br/>

<a NAME="E4:102"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
 the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon" title="c4">A2</font></span>)</span> <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="c3">A1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a>  the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c4">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:102"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="circcomb.html#D4" target="_self" title="CIRCCOMB:def.4">Def4</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="circcomb.html#D10" target="_self" title="CIRCCOMB:def.10">CIRCCOMB:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S1</font> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <font color="Maroon" title="c2">S2</font></span>)</span> implies <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon" title="c4">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">g</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<a NAME="E5:102"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">A1</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">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E6:102"/><span class="lab"><font color="Green" title="E59">A6</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c4">A2</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="c2">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E7:102"/><span class="lab"><font color="Green" title="E60">A7</font></span>: 
 the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S1</font> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <font color="Maroon" title="c2">S2</font></span>)</span> <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">S1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circcomb.html#D2" target="_self" title="CIRCCOMB:def.2">Def2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:102"/>
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S1</font> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <font color="Maroon" title="c2">S2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon" title="c4">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">g</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<a NAME="E9:102"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A8</font></span>: 
( <font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S1</font> or <font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">S2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:102"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E10:102"/>
 the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">A1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c4">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E2:102"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="circcomb.html#T48" target="_self" title="CIRCCOMB:th.48">Th56</a></span>;<br/>
<a NAME="E11:102"/><span class="kw">then </span>
( (  the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon" title="c4">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">g</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="c3">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">g</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">g</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> ) or (  the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon" title="c4">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">g</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="c4">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">g</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c4">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">g</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E2:102"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E4:102"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E5:102"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="txt" href="#E6:102"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E9:102"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="ref" href="circcomb.html#D10" target="_self" title="CIRCCOMB:def.10">Def10</a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a>, <a class="ref" href="funct_4.html#T15" title="FUNCT_4:th.15">FUNCT_4:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:102"/>
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon" title="c4">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">g</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
