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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="altcat_1.html#L2" target="_self">AltCatStr</a> ;<br/>

<b>assume </b><a NAME="E1:48_2_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="altcat_1.html#V12" target="_self">with_units</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="altcat_1.html#V10" target="_self">pseudo-functional</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="altcat_1.html#V2" target="_self">transitive</a> )
 ;<br/>

<b>thus </b><a NAME="E2:48_2_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="altcat_1.html#V8" target="_self">quasi-functional</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1_1"><b>proof </b></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="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D13" target="_self">ALTCAT_1:def 13</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_2_1_1_1"/>
( <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_2_1_1_1_1"/>
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_2_1_1_1_1"/>
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_2_1_1_1_2"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:48_2_1_1_1_2"/>
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="altcat_1.html#E1:48_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_1.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E3:48_2_1_1_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="altcat_1.html#E1:48_2_1_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><a NAME="E4:48_2_1_1_1_2"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="altcat_1.html#E1:48_2_1_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:48_2_1_1_1_2"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E6:48_2_1_1_1_2"/>
the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="altcat_1.html#E1:48_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_1.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E7:48_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="altcat_1.html#E4:48_2_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="altcat_1.html#E5:48_2_1_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/><b>hence </b><a NAME="E8:48_2_1_1_1_2"/>
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</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="altcat_1.html#E3:48_2_1_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zfmisc_1.html#T138">ZFMISC_1:138</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D14" target="_self">ALTCAT_1:def 14</a><br/>

<b>thus </b><a NAME="E3:48_2_1"/>
( <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">b<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="altcat_1.html#E1:48_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="altcat_1.html#T18" target="_self">Th18</a></i>;<br/>


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