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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>;<br/>

<a NAME="E1:70"/><i><font color="Green" title="E41">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <span class="p2">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span></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 <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="altcat_4.html#D5" target="_self" title="ALTCAT_4:def.5">Def5</a></i>;<br/>
<a NAME="E2:70"/><i><font color="Green" title="E42">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</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="c1">C</font>
 
<b>by </b><i><a class="ref" href="altcat_4.html#D5" target="_self" title="ALTCAT_4:def.5">Def5</a></i>;<br/>
<a NAME="E3:70"/><i><font color="Green" title="E43">A3</font></i>: 
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <span class="p2">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="altcat_4.html#D5" target="_self" title="ALTCAT_4:def.5">Def5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:70_1"/><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c4">o2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:70_1"/><i><font color="Green" title="E45">A5</font></i>: 
( <font color="Maroon" title="c3">o1</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="c1">C</font> &amp; <font color="Maroon" title="c4">o2</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="c1">C</font> &amp; <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">o1</font> = <font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c6">o2</font> = <font color="Maroon" title="c4">o2</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span> <b>by </b><i><a class="txt" href="altcat_4.html#E3:70_1"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="altcat_4.html#D5" target="_self" title="ALTCAT_4:def.5">Def5</a></i>;<br/><b>thus </b><a NAME="E5:70_1"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <span class="p2">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:70_1_1"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <span class="p2">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E1:70"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="altcat_4.html#E2:70"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="altcat_4.html#E3:70"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="altcat_4.html#E1:70_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="altcat_2.html#D2" title="ALTCAT_2:def.2">ALTCAT_2:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c7">n</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:70_1_1"/>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
 ;<br/>

<a NAME="E3:70_1_1"/><b>then </b><i><font color="Green" title="E46">A6</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">o1</font>,<font color="Maroon" title="c6">o2</font>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E3:70_1"><i><font color="Green" title="E45">A5</font></i></a></i>;<br/>
<a NAME="E4:70_1_1"/><b>then </b><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">o1</font>,<font color="Maroon" title="c6">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
;<br/>
<b>reconsider </b><font color="Maroon" title="c8">n1</font> = <font color="Maroon" title="c7">n</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">o1</font>,<font color="Maroon" title="c6">o2</font> <b>by </b><i><a class="txt" href="altcat_4.html#E3:70_1_1"><i><font color="Green" title="E46">A6</font></i></a></i>;<br/>
<a NAME="E6:70_1_1"/><i><font color="Green" title="E48">A8</font></i>: 
<font color="Maroon" title="c8">n1</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">o2</font>,<font color="Maroon" title="c5">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E4:70_1_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="altcat_4.html#T52" target="_self" title="ALTCAT_4:th.52">Th52</a></i>;<br/>
<a NAME="E7:70_1_1"/>
<font color="Maroon" title="c8">n1</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E4:70_1_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="altcat_4.html#T52" target="_self" title="ALTCAT_4:th.52">Th52</a></i>;<br/>
<a NAME="E8:70_1_1"/><b>then </b>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <span class="p2">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">o1</font>,<font color="Maroon" title="c6">o2</font>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E4:70_1_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="txt" href="altcat_4.html#E6:70_1_1"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="altcat_4.html#D5" target="_self" title="ALTCAT_4:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E9:70_1_1"/>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <span class="p2">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E3:70_1"><i><font color="Green" title="E45">A5</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:70"/><b>then </b>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <span class="p2">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E1:70"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="altcat_4.html#E2:70"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>
<b>hence </b><a NAME="E6:70"/>
 <a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E1:70"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="altcat_2.html#T26" title="ALTCAT_2:th.26">ALTCAT_2:26</a></i>;<br/>


</div>
