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<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="prob_2.html#D3">PROB_2:def 3</a>,<a class="ref" href="msafree1.html#D2">MSAFREE1:def 2</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:73_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:73_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:73_1_1_1"/>
( the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:73_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:73_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="catalg_1.html#T24" target="_self">Th24</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:73_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="catalg_1.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E10:73_1_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:73_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="catalg_1.html#E7:73_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="catalg_1.html#E9:73_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="catalg_1.html#D15" target="_self">Def15</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E12:73_1_1_1"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E10:73_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><b>hence </b><a NAME="E13:73_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="catalg_1.html#E1:73_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="catalg_1.html#E7:73_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="catalg_1.html#E9:73_1_1_1"><i><font color="Green">E43</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_6.html#D1">MSUALG_6:def 1</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_1_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="ref" href="catalg_1.html#T25" target="_self">Th25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_2_1"><b>suppose </b></a><a NAME="E1:73_1_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:73_1_1_2_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="catalg_1.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E4:73_1_1_2_1"/>
(  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:73_1_1_2_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="catalg_1.html#T36" target="_self">Th36</a>, <a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/><b>hence </b><a NAME="E5:73_1_1_2_1"/>
(  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  not  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_2_2"><b>suppose </b></a><a NAME="E1:73_1_1_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:73_1_1_2_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="catalg_1.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E4:73_1_1_2_2"/><i><font color="Green">E41</font></i>: 
(  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp;  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:73_1_1_2_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="catalg_1.html#T37" target="_self">Th37</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>assume </b><a NAME="E6:73_1_1_2_2"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E7:73_1_1_2_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:73_1_1_2_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_6.html#T2">FUNCT_6:2</a></i>;<br/><b>hence </b><a NAME="E8:73_1_1_2_2"/>
not  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:73_1_1_2_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
