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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S1</font> be    <a href="circcmb3.html#M1" target="_self">Signature</a> of <font color="Maroon">X</font>, <font color="Maroon">S2</font> be    <a href="circcmb3.html#M1" target="_self">Signature</a> of <font color="Maroon">X</font>;<br/>



<a NAME="E1:61"/><i><font color="Green">E34</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">S2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#D2">CIRCCOMB:def 2</a></i>;<br/>
<b>let </b><font color="Maroon">A1</font> be    <a href="circcmb3.html#M2" target="_self">Circuit</a> of <font color="Maroon">X</font>,<font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">A2</font> be    <a href="circcmb3.html#M2" target="_self">Circuit</a> of <font color="Maroon">X</font>,<font color="Maroon">S2</font>;<br/>



<a NAME="E2:61"/>
<font color="Maroon">A1</font> <a href="circcomb.html#R2">tolerates</a> <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="circcmb3.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E3:61"/><b>then </b><i><font color="Green">E36</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="partfun1.html#R1">tolerates</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#D3">CIRCCOMB:def 3</a></i>;<br/>
<a NAME="E4:61"/><b>then </b><i><font color="Green">E37</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="circcomb.html#K4">+*</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="circcomb.html#K1">+*</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#D4">CIRCCOMB:def 4</a></i>;<br/>
<a NAME="E5:61"/><i><font color="Green">E38</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:61"/><i><font color="Green">E45</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> is <a href="finset_1.html#V2">locally-finite</a> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font> is <a href="finset_1.html#V2">locally-finite</a> )
 
<b>by </b><i><a class="ref" href="msafree2.html#D11">MSAFREE2:def 11</a></i>;<br/>
<a NAME="E7:61"/>
<font color="Maroon">A1</font> <a href="circcomb.html#K4">+*</a> <font color="Maroon">A2</font> is <a href="msafree2.html#V4">locally-finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_circ.html#D3">PRE_CIRC:def 3</a>,<a class="ref" href="msafree2.html#D11">MSAFREE2:def 11</a><br/>

<b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">S2</font></span>)</span>
 ;<br/>

<a NAME="E2:61_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> or <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:61_1"/><b>then </b>
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="circcomb.html#K4">+*</a> <font color="Maroon">A2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="circcomb.html#K4">+*</a> <font color="Maroon">A2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> ) )
 
<b>by </b><i>, , , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a>, <a class="ref" href="funct_4.html#T16">FUNCT_4:16</a></i>;<br/>
<b>hence </b><a NAME="E4:61_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="circcomb.html#K4">+*</a> <font color="Maroon">A2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="circcmb3.html#E1:61"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pre_circ.html#D3">PRE_CIRC:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:61"/>
<font color="Maroon">A1</font> <a href="circcomb.html#K4">+*</a> <font color="Maroon">A2</font> is   <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">S1</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">S2</font>
 ;<br/>


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