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<div class="add">

<span class="kw">set </span><font color="Maroon" title="c1">alt</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<span class="kw">set </span><font color="Maroon" title="c2">IT</font> =  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<span class="kw">set </span><font color="Maroon" title="c3">I</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<span class="kw">set </span><font color="Maroon" title="c4">G</font> =  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="altcat_1.html#D7" onmouseover="rs('altcat_1/D7')" onmouseout="rh()">ALTCAT_1:def 7</a>,<a class="ref" href="altcat_1.html#D16" onmouseover="rs('altcat_1/D16')" onmouseout="rh()">ALTCAT_1:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() is  <a href="altcat_1.html#V4" title="ALTCAT_1:attr.4">with_right_units</a> </span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c5">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font>) &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b3">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b3">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">i</font>,<font color="Maroon" title="c5">j</font>) holds <br/><span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Olive" title="b2">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">j9</font> = <font color="Maroon" title="c5">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><span class="kw">consider </span><font color="Maroon" title="c7">f</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:3_1_1"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="tarski.html#R2" title="TARSKI: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">j9</font>,<font color="Maroon" title="c6">j9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E4:3_1_1"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
 for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI: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="Olive" title="b1">b</font>,<font color="Maroon" title="c6">j9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> holds <br/><font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c6">j9</font>,<font color="Maroon" title="c6">j9</font>,<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E3">A3</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">f</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font>) &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">j</font>) holds <br/><span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">f</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) )</span><br/><span class="kw">thus </span><a NAME="E5:3_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">j</font>) holds <br/><span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">f</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>) holds <br/><span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">f</font>,<font color="Olive" title="b1">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c10">g</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>) implies <span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c10">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> )</span><br/><span class="kw">assume </span><a NAME="E6:3_1_1"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c10">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">i9</font> = <font color="Maroon" title="c9">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><a NAME="E8:3_1_1"/>
 the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c11">i9</font>,<font color="Maroon" title="c6">j9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><a NAME="E9:3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A7</font></span>: 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c8">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="txt" href="#E6:3_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">f9</font> = <font color="Maroon" title="c8">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">j9</font>,<font color="Maroon" title="c6">j9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">g9</font> = <font color="Maroon" title="c10">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">i9</font>,<font color="Maroon" title="c6">j9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:3_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c10">g</font>) = 
<font color="Maroon" title="c12">f9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c13">g9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E6:3_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="ref" href="altcat_1.html#D8" onmouseover="rs('altcat_1/D8')" onmouseout="rh()">ALTCAT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c10">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="txt" href="#E3:3_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E6:3_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E9:3_1_1"><span class="lab"><font color="Green" title="E7">A7</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="altcat_1.html#D6" onmouseover="rs('altcat_1/D6')" onmouseout="rh()">ALTCAT_1:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>) &amp; (  for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive">b<sub>3</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Olive">b<sub>2</sub></font>) or <span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Olive">b<sub>2</sub></font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> ) ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">i9</font> = <font color="Maroon" title="c5">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E4:3_1"/><span class="lab"><font color="Green" title="E4">A8</font></span>: 
<font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI: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">i9</font>,<font color="Maroon" title="c6">i9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E5:3_1"/><span class="lab"><font color="Green" title="E5">A9</font></span>: 
 for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">g</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">g</font> <a href="tarski.html#R2" title="TARSKI: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">i9</font>,<font color="Olive" title="b1">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> holds <br/><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c7">e</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E2">A2</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">e</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>) &amp; (  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Olive">b<sub>1</sub></font>) or <span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Olive">b<sub>1</sub></font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E6:3_1"/>
<font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1"><span class="lab"><font color="Green" title="E4">A8</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive">b<sub>2</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Olive">b<sub>1</sub></font>) or <span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Olive">b<sub>1</sub></font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c7">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">e9</font> = <font color="Maroon" title="c7">e</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">i9</font>,<font color="Maroon" title="c6">i9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1"><span class="lab"><font color="Green" title="E4">A8</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font>) or <span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c7">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c10">f</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font>) or <span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c7">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> )</span><br/>



<span class="kw">assume </span><a NAME="E8:3_1"/><span class="lab"><font color="Green" title="E6">A10</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c7">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">j9</font> = <font color="Maroon" title="c9">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E10:3_1"/>
 the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">i9</font>,<font color="Maroon" title="c11">j9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
;<br/>
<a NAME="E11:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A11</font></span>: 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c7">e</font>,<font color="Maroon" title="c10">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3_1"><span class="lab"><font color="Green" title="E5">A9</font></span></a>, <a class="txt" href="#E8:3_1"><span class="lab"><font color="Green" title="E6">A10</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">f9</font> = <font color="Maroon" title="c10">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">i9</font>,<font color="Maroon" title="c11">j9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:3_1"><span class="lab"><font color="Green" title="E6">A10</font></span></a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c7">e</font>) = 
<font color="Maroon" title="c12">f9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c8">e9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1"><span class="lab"><font color="Green" title="E4">A8</font></span></a>, <a class="txt" href="#E8:3_1"><span class="lab"><font color="Green" title="E6">A10</font></span></a>, <a class="ref" href="altcat_1.html#D8" onmouseover="rs('altcat_1/D8')" onmouseout="rh()">ALTCAT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c10">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="txt" href="#E4:3_1"><span class="lab"><font color="Green" title="E4">A8</font></span></a>, <a class="txt" href="#E8:3_1"><span class="lab"><font color="Green" title="E6">A10</font></span></a>, <a class="txt" href="#E11:3_1"><span class="lab"><font color="Green" title="E7">A11</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
