<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="catalg_1.html#NM1">Signature</a>;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:56"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is    <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is    <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#R1">tolerates</a> <font color="Maroon">c<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is    <a href="instalg1.html#M1">Subsignature</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> is    <a href="instalg1.html#M1">Subsignature</a> of <font color="Maroon">c<sub>3</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="algspec1.html#D5" target="_self">ALGSPEC1:def 5</a><br/>

<a NAME="E2:56_1"/><b>then </b>
( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="instalg1.html#T12">INSTALG1:12</a></i>;<br/>
<b>hence </b><a NAME="E3:56_1"/>
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#R1">tolerates</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="partfun1.html#T131">PARTFUN1:131</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="circcomb.html#D1">CIRCCOMB:def 1</a><br/>

<a NAME="E4:56_1"/>
( the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="algspec1.html#E1:56_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="instalg1.html#T12">INSTALG1:12</a></i>;<br/>
<b>hence </b><a NAME="E5:56_1"/>
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#R1">tolerates</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="partfun1.html#T131">PARTFUN1:131</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:56"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:56"/><b>then </b>
(  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_4.html#T4">FUNCT_4:4</a></i>;<br/>
<a NAME="E4:56"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#R1">tolerates</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T131">PARTFUN1:131</a></i>;<br/>
<a NAME="E5:56"/><i><font color="Green">E48</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="circcomb.html#D2">CIRCCOMB:def 2</a></i>;<br/>
<a NAME="E6:56"/>
( <font color="Maroon">c<sub>3</sub></font> is    <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is    <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> is    <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_2"/>
(  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="instalg1.html#D2">INSTALG1:def 2</a>,<a class="ref" href="algspec1.html#D5" target="_self">ALGSPEC1:def 5</a><br/>

<a NAME="E2:56_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="algspec1.html#E4:56"><i><font color="Green">E47</font></i></a>, <a class="ref" href="algspec1.html#T51" target="_self">Th51</a></i>;<br/>
<a NAME="E3:56_2"/><b>then </b>
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="algspec1.html#E5:56"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_4.html#T23">FUNCT_4:23</a></i>;<br/>
<b>hence </b><a NAME="E4:56_2"/>
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>, <a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="algspec1.html#E5:56"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_4.html#T23">FUNCT_4:23</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="instalg1.html#D2">INSTALG1:def 2</a>,<a class="ref" href="algspec1.html#D5" target="_self">ALGSPEC1:def 5</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:56"/>
( <font color="Maroon">c<sub>3</sub></font> is    <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is    <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">c<sub>2</sub></font> implies ( <font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#R1">tolerates</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is    <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="algspec1.html#E1:56"><i><font color="Green">E46</font></i></a></i>;<br/>

<b>assume </b><a NAME="E8:56"/>
<font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#R1">tolerates</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E9:56"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="algspec1.html#T49" target="_self">Th49</a>, <a class="ref" href="algspec1.html#T50" target="_self">Th50</a></i>;<br/>
<b>hence </b><a NAME="E10:56"/>
(  not <font color="Maroon">c<sub>3</sub></font> is    <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> is    <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is    <a href="algspec1.html#M2" target="_self">Extension</a> of <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="algspec1.html#T48" target="_self">Th48</a></i>;<br/>


</div>
