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<b>let </b><font color="Maroon">it1</font> be    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font>, <font color="Maroon">it2</font> be    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font>;<br/>

<b>assume </b><a NAME="E1:2_1_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font> being  <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">it1</font> holds <br/><font color="Maroon">it1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">SCS</font>,<font color="Olive">x</font>
 ;<br/>

<b>assume </b><a NAME="E2:2_1_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">x</font> being  <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">it2</font> holds <br/><font color="Maroon">it2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">SCS</font>,<font color="Olive">x</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:2_1_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font>
 ;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">i</font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font> ;<br/><a NAME="E3:2_1_2_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">it2</font> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> )
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:2_1_2_1"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">it1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">SCS</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">it2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">SCS</font>,<font color="Maroon">v</font> )
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:2_1_2_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp;  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">SCS</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> :  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">SCS</font></span>)</span> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="msualg_2.html#D4">MSUALG_2:def 4</a></i>;<br/><b>consider </b><font color="Maroon">a1</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E8:2_1_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">it1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 <b>and </b><br/><a NAME="E9:2_1_2_1"/><i><font color="Green">E56</font></i>: 
 ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">SCS</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">a2</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font><b> such that </b><br/><a NAME="E11:2_1_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">it2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 <b>and </b><br/><a NAME="E12:2_1_2_1"/><i><font color="Green">E59</font></i>: 
 ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">SCS</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">o1</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font><b> such that </b><br/><a NAME="E14:2_1_2_1"/><i><font color="Green">E61</font></i>: 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E15:2_1_2_1"/><i><font color="Green">E62</font></i>: 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E16:2_1_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">SCS</font></span>)</span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">o2</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font><b> such that </b><br/><a NAME="E18:2_1_2_1"/>
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E19:2_1_2_1"/><i><font color="Green">E65</font></i>: 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E20:2_1_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">SCS</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E21:2_1_2_1"/><i><font color="Green">E67</font></i>: 
(  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, , <a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>;<br/><a NAME="E22:2_1_2_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="ref" href="msafree2.html#D6">MSAFREE2:def 6</a></i>;<br/><a NAME="E23:2_1_2_1"/><i><font color="Green">E69</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">IIG</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">IIG</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E24:2_1_2_1"/><i><font color="Green">E70</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="lang1.html#K2">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font>
 ;<br/><i><font color="Green">E71</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">SCS</font></span>)</span> = 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o1</font>,<font color="Maroon">SCS</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">IIG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, , <a class="ref" href="pre_circ.html#T5">PRE_CIRC:5</a></i>
;<br/><b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E27:2_1_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">SCS</font></span>)</span>
 <b>and </b><br/><a NAME="E28:2_1_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o1</font>,<font color="Maroon">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E30:2_1_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">SCS</font></span>)</span>
 <b>and </b><br/><a NAME="E31:2_1_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o2</font>,<font color="Maroon">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E32:2_1_2_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E33:2_1_2_1"/>
<font color="Maroon">it1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">it2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , , , <a class="ref" href="msafree2.html#D6">MSAFREE2:def 6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:2_1_2"/>
<font color="Maroon">it1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">it2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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