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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">S1</font>, <font color="Olive" title="b2">S2</font> being   <a href="circcmb3.html#M1" title="CIRCCMB3:mode.1">Signature</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">A1</font> being   <a href="circcmb3.html#M2" title="CIRCCMB3:mode.2">Circuit</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">S1</font><br/> for <font color="Olive" title="b4">A2</font> being   <a href="circcmb3.html#M2" title="CIRCCMB3:mode.2">Circuit</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">S2</font> holds  <font color="Olive" title="b3">A1</font> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">+*</a> <font color="Olive" title="b4">A2</font> is   <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Olive" title="b1">S1</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Olive" title="b2">S2</font></span><br/><b>let </b><font color="Maroon" title="c2">S1</font>, <font color="Maroon" title="c3">S2</font> be    <a href="circcmb3.html#M1" title="CIRCCMB3:mode.1">Signature</a> of <font color="Maroon" title="c1">X</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">A1</font> being   <a href="circcmb3.html#M2" title="CIRCCMB3:mode.2">Circuit</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S1</font><br/> for <font color="Olive" title="b2">A2</font> being   <a href="circcmb3.html#M2" title="CIRCCMB3:mode.2">Circuit</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">S2</font> holds  <font color="Olive" title="b1">A1</font> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">+*</a> <font color="Olive" title="b2">A2</font> is   <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">S1</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon" title="c3">S2</font></span><br/>



<a NAME="E1:61"/><i><font color="Green" title="E30">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon" title="c3">S2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S2</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#D2" title="CIRCCOMB:def.2">CIRCCOMB:def 2</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">A1</font> be    <a href="circcmb3.html#M2" title="CIRCCMB3:mode.2">Circuit</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S1</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">A2</font> being   <a href="circcmb3.html#M2" title="CIRCCMB3:mode.2">Circuit</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">S2</font> holds  <font color="Maroon" title="c4">A1</font> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">+*</a> <font color="Olive" title="b1">A2</font> is   <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">S1</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon" title="c3">S2</font></span><br/><b>let </b><font color="Maroon" title="c5">A2</font> be    <a href="circcmb3.html#M2" title="CIRCCMB3:mode.2">Circuit</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">S2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A1</font> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">+*</a> <font color="Maroon" title="c5">A2</font> is   <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">S1</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon" title="c3">S2</font></span><br/>



<a NAME="E2:61"/>
<font color="Maroon" title="c4">A1</font> <a href="circcomb.html#R2" title="CIRCCOMB:pred.2">tolerates</a> <font color="Maroon" title="c5">A2</font>
 
<b>by </b><i><a class="ref" href="circcmb3.html#T30" target="_self" title="CIRCCMB3:th.30">Th30</a></i>;<br/>
<a NAME="E3:61"/><b>then </b><i><font color="Green" title="E31">A2</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">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="c5">A2</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#D3" title="CIRCCOMB:def.3">CIRCCOMB:def 3</a></i>;<br/>
<a NAME="E4:61"/><b>then </b><i><font color="Green" title="E32">A3</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">+*</a> <font color="Maroon" title="c5">A2</font></span>)</span> <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="c4">A1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">A2</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#D4" title="CIRCCOMB:def.4">CIRCCOMB:def 4</a></i>;<br/>
<a NAME="E5:61"/><i><font color="Green" title="E33">A4</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S1</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S2</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:61"/><i><font color="Green" title="E34">A5</font></i>: 
( the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">A1</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> &amp; the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">A2</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> )
 
<b>by </b><i><a class="ref" href="msafree2.html#D11" title="MSAFREE2:def.11">MSAFREE2:def 11</a></i>;<br/>
<a NAME="E7:61"/>
<font color="Maroon" title="c4">A1</font> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">+*</a> <font color="Maroon" title="c5">A2</font> is <a href="msafree2.html#V4" title="MSAFREE2:attr.4">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" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_circ.html#D3" title="PRE_CIRC:def.3">PRE_CIRC:def 3</a>,<a class="ref" href="msafree2.html#D11" title="MSAFREE2:def.11">MSAFREE2:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon" title="c3">S2</font></span>)</span> or the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">+*</a> <font color="Maroon" title="c5">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/>

<b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon" title="c3">S2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">+*</a> <font color="Maroon" title="c5">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/>

<a NAME="E2:61_1"/><b>then </b>
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S1</font> or <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S2</font> )
 
<b>by </b><i><a class="txt" href="#E1:61"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:61_1"/><b>then </b>
( ( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S1</font> &amp; the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">+*</a> <font color="Maroon" title="c5">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</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="c4">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> ) or ( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S2</font> &amp; the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">+*</a> <font color="Maroon" title="c5">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</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="c5">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> ) )
 
<b>by </b><i><a class="txt" href="#E3:61"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="#E4:61"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="#E5:61"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a>, <a class="ref" href="funct_4.html#T16" title="FUNCT_4:th.16">FUNCT_4:16</a></i>;<br/>
<b>hence </b><a NAME="E4:61_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">A1</font> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">+*</a> <font color="Maroon" title="c5">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="#E6:61"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="pre_circ.html#D3" title="PRE_CIRC:def.3">PRE_CIRC:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:61"/>
<font color="Maroon" title="c4">A1</font> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">+*</a> <font color="Maroon" title="c5">A2</font> is   <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">S1</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon" title="c3">S2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
