<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">C</font> is  <a href="cat_4.html#V1" title="CAT_4:attr.1">with_finite_product</a>  iff (  ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> st <font color="Olive" title="b1">a</font> is  <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a>  &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:3"/>
( <font color="Maroon" title="c1">C</font> is  <a href="cat_4.html#V1" title="CAT_4:attr.1">with_finite_product</a>  implies (  ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> st <font color="Olive" title="b1">a</font> is  <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a>  &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font> ) ) ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> st <font color="Olive" title="b1">a</font> is  <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a>  &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font> ) ) implies <font color="Maroon" title="c1">C</font> is  <a href="cat_4.html#V1" title="CAT_4:attr.1">with_finite_product</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c2">F</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T25" title="RELSET_1:th.25">RELSET_1:25</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:3_1"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
for <font color="Olive" title="b1">I</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">I</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">I</font> st <br/>(  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b4">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b4">F9</font> )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="cat_4.html#D3" target="_self" title="CAT_4:def.3">CAT_4:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> st <font color="Olive" title="b1">a</font> is  <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a>  &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font> ) ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c4">F9</font> being    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c3">a</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <span class="kw"> such that </span><br/><a NAME="E4:3_1"/>
 <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c4">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">and </span><br/><a NAME="E5:3_1"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c4">F9</font>
 ;<br/>
<a NAME="E6:3_1"/>
<font color="Maroon" title="c3">a</font> is  <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="ref" href="cat_3.html#T55" title="CAT_3:th.55">CAT_3:55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:3_1"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> st <font color="Olive" title="b1">a</font> is  <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Olive" title="b1">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">F</font> = <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c9">F9</font> being    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c8">c</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span><span class="kw"> such that </span><br/><a NAME="E9:3_1"/><span class="lab"><font color="Green" title="E4">A3</font></span>: 
 <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c9">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
 <span class="kw">and </span><br/><a NAME="E10:3_1"/><span class="lab"><font color="Green" title="E5">A4</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c9">F9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E2">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c8">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c10">p1</font> = <font color="Maroon" title="c9">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c8">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c10">p1</font>,<font color="Olive" title="b1">p2</font> )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c11">p2</font> = <font color="Maroon" title="c9">F9</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c8">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font> )</span><br/>

<a NAME="E11:3_1"/><span class="lab"><font color="Green" title="E6">A5</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E12:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A6</font></span>: 
<font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>;<br/>
<a NAME="E13:3_1"/><span class="lab"><font color="Green" title="E8">A7</font></span>: 
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:3_1"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:3_1"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="ref" href="cat_3.html#T45" title="CAT_3:th.45">CAT_3:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c8">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font> )</span><br/>

<a NAME="E15:3_1"/>
( <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T7" title="CAT_3:th.7">CAT_3:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:3_1"/>
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:3_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E13:3_1"><span class="lab"><font color="Green" title="E8">A7</font></span></a>, <a class="txt" href="#E11:3_1"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">CAT_3:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font></span><br/>

<a NAME="E17:3_1"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">F9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:3_1"><span class="lab"><font color="Green" title="E8">A7</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E18:3_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:3_1"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="ref" href="funct_4.html#T69" title="FUNCT_4:th.69">FUNCT_4:69</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:3_1"/>
<font color="Maroon" title="c8">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:3_1"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="ref" href="cat_3.html#T59" title="CAT_3:th.59">CAT_3:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c2">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><a NAME="E3:3"/><span class="lab"><font color="Green" title="E2">A8</font></span>: 
<font color="Maroon" title="c2">a</font> is  <a href="cat_1.html#V6" title="CAT_1:attr.6">terminal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> st <br/>for <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  holds <br/>(  not  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> or  not  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> or  not  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> or  not <font color="Olive" title="b3">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font> ) or <font color="Maroon" title="c1">C</font> is  <a href="cat_4.html#V1" title="CAT_4:attr.1">with_finite_product</a>  )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">I</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">I</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> ex <font color="Olive" title="b3">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">I</font> st <br/>(  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b4">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b4">F9</font> );<br/>
<span class="kw">assume </span><a NAME="E4:3"/><span class="lab"><font color="Green" title="E3">A9</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  ex <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b3">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> is  <a href="cat_4.html#V1" title="CAT_4:attr.1">with_finite_product</a> </span><br/>

<a NAME="E5:3"/><span class="lab"><font color="Green" title="E4">A10</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:3_2"/><span class="lab"><font color="Green" title="E5">A11</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">I</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">I</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> ex <font color="Olive" title="b2">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b3">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b2">a</font>,<font color="Maroon" title="c4">I</font> st <br/>(  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b3">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b3">F9</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">I</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">I</font> st <br/>(  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b2">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Olive" title="b1">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b2">F9</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E2:3_2"/><span class="lab"><font color="Green" title="E6">A12</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">I</font> st <br/>(  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b2">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Olive" title="b1">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b2">F9</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">I</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">Ix</font> = <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">sx</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">G</font> = <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></span>;<br/>
<a NAME="E7:3_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_2"><span class="lab"><font color="Green" title="E6">A12</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<span class="kw">then </span> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">Ix</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">sx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></span>
<br/>.= 
<font color="Maroon" title="c3">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_2"><span class="lab"><font color="Green" title="E6">A12</font></span></a>, <a class="ref" href="xcmplx_1.html#T26" title="XCMPLX_1:th.26">XCMPLX_1:26</a></span>
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c11">G9</font> being    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span><span class="kw"> such that </span><br/><a NAME="E10:3_2"/><span class="lab"><font color="Green" title="E7">A13</font></span>: 
 <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c11">G9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">G</font>
 <span class="kw">and </span><br/><a NAME="E11:3_2"/><span class="lab"><font color="Green" title="E8">A14</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c11">G9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2"><span class="lab"><font color="Green" title="E5">A11</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c13">p1</font>, <font color="Maroon" title="c14">p2</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E13:3_2"/><span class="lab"><font color="Green" title="E9">A15</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c13">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</font>
 <span class="kw">and </span><br/><a NAME="E14:3_2"/><span class="lab"><font color="Green" title="E10">A16</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c14">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</font>
 <span class="kw">and </span><br/><a NAME="E15:3_2"/><span class="lab"><font color="Green" title="E11">A17</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">and </span><br/><a NAME="E16:3_2"/><span class="lab"><font color="Green" title="E12">A18</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c14">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">and </span><br/><a NAME="E17:3_2"/><span class="lab"><font color="Green" title="E13">A19</font></span>: 
<font color="Maroon" title="c12">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c13">p1</font>,<font color="Maroon" title="c14">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3"><span class="lab"><font color="Green" title="E3">A9</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">F9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span>;<br/>
<a NAME="E18:3_2"/><span class="lab"><font color="Green" title="E14">A20</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E19:3_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></span>;<br/>
<a NAME="E20:3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A21</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E21:3_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E16">A22</font></span>: <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c4">I</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:3_2"><span class="lab"><font color="Green" title="E14">A20</font></span></a>, <a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>
<br/>.= 
<font color="Maroon" title="c4">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_2"><span class="lab"><font color="Green" title="E6">A12</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></span>
;<br/>
<a NAME="E23:3_2"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">F9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">I</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:3_2"><span class="lab"><font color="Green" title="E15">A21</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/>
<a NAME="E25:3_2"/><span class="lab"><font color="Green" title="E17">A23</font></span>: 
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c11">G9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c10">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D14" title="CAT_3:def.14">CAT_3:def 14</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> implies <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c16">F9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:3_2_3"/><span class="lab"><font color="Green" title="E18">A24</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c16">F9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:3_2_3_1_1"/>
( <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> or <font color="Maroon" title="c17">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:3_2_3_1_1_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c16">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">y</font></span><br/><div class="add"><a NAME="E2:3_2_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A25</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><font color="Maroon" title="c16">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">y</font> = 
<font color="Maroon" title="c16">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_3"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:3_2"><span class="lab"><font color="Green" title="E14">A20</font></span></a>, <a class="txt" href="#E2:3_2_3_1_1_1"><span class="lab"><font color="Green" title="E19">A25</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
<font color="Maroon" title="c14">p2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_2_3_1_1_1"><span class="lab"><font color="Green" title="E19">A25</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> = 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:3_2"><span class="lab"><font color="Green" title="E10">A16</font></span></a>, <a class="txt" href="#E1:3_2_3"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c16">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_3"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="ref" href="cat_3.html#D3" title="CAT_3:def.3">CAT_3:def 3</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:3_2_3_1_1_2"/><span class="lab"><font color="Green" title="E19">A26</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c16">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font></span><br/><div class="add"><a NAME="E2:3_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A27</font></span>: 
not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:3_2_3_1_1_2"/><span class="lab"><font color="Green" title="E21">A28</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_3"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="txt" href="#E1:3_2_3_1_1_2"><span class="lab"><font color="Green" title="E19">A26</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/><font color="Maroon" title="c16">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">y</font> = 
<font color="Maroon" title="c16">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_3"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:3_2"><span class="lab"><font color="Green" title="E14">A20</font></span></a>, <a class="txt" href="#E22:3_2"><span class="lab"><font color="Green" title="E16">A22</font></span></a>, <a class="txt" href="#E1:3_2_3"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="txt" href="#E2:3_2_3_1_1_2"><span class="lab"><font color="Green" title="E20">A27</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_2_3_1_1_2"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c16">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">y</font> = 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_3"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="ref" href="cat_3.html#D3" title="CAT_3:def.3">CAT_3:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_2_3_1_1_2"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="ref" href="cat_3.html#D3" title="CAT_3:def.3">CAT_3:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:3_2"><span class="lab"><font color="Green" title="E9">A15</font></span></a>, <a class="txt" href="#E15:3_2"><span class="lab"><font color="Green" title="E11">A17</font></span></a>, <a class="txt" href="#E25:3_2"><span class="lab"><font color="Green" title="E17">A23</font></span></a>, <a class="ref" href="cat_3.html#T20" title="CAT_3:th.20">CAT_3:20</a></span>
<br/>.= 
<font color="Maroon" title="c12">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_2_3_1_1_2"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="ref" href="cat_3.html#T2" title="CAT_3:th.2">CAT_3:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_3"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:3_2_3"/>
<span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c16">F9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_3"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E27:3_2"/><span class="kw">then </span>
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c16">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c12">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">F9</font> = <font color="Maroon" title="c16">F9</font> as    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c12">c</font>,<font color="Maroon" title="c4">I</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D14" title="CAT_3:def.14">CAT_3:def 14</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c12">c</font>,<font color="Maroon" title="c4">I</font> st <br/>(  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b1">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c12">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b1">F9</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c17">F9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c12">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c17">F9</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> implies <span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:3_2_4"/><span class="lab"><font color="Green" title="E18">A29</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:3_2_4_1_1"/>
( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:3_2_4_1_1_1"/><span class="lab"><font color="Green" title="E19">A30</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font></span><br/><div class="add"><a NAME="E2:3_2_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A31</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font> = 
<font color="Maroon" title="c17">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_4"><span class="lab"><font color="Green" title="E18">A29</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:3_2"><span class="lab"><font color="Green" title="E14">A20</font></span></a>, <a class="txt" href="#E2:3_2_4_1_1_1"><span class="lab"><font color="Green" title="E20">A31</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
<font color="Maroon" title="c14">p2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_2_4_1_1_1"><span class="lab"><font color="Green" title="E20">A31</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:3_2_4_1_1_1"/>
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:3_2"><span class="lab"><font color="Green" title="E12">A18</font></span></a>, <a class="txt" href="#E1:3_2_4"><span class="lab"><font color="Green" title="E18">A29</font></span></a>, <a class="txt" href="#E1:3_2_4_1_1_1"><span class="lab"><font color="Green" title="E19">A30</font></span></a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">CAT_3:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:3_2_4_1_1_2"/><span class="lab"><font color="Green" title="E19">A32</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font></span><br/><div class="add"><a NAME="E2:3_2_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A33</font></span>: 
not <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:3_2_4_1_1_2"/><span class="lab"><font color="Green" title="E21">A34</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_4"><span class="lab"><font color="Green" title="E18">A29</font></span></a>, <a class="txt" href="#E1:3_2_4_1_1_2"><span class="lab"><font color="Green" title="E19">A32</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font> = 
<font color="Maroon" title="c17">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_4"><span class="lab"><font color="Green" title="E18">A29</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:3_2"><span class="lab"><font color="Green" title="E14">A20</font></span></a>, <a class="txt" href="#E22:3_2"><span class="lab"><font color="Green" title="E16">A22</font></span></a>, <a class="txt" href="#E1:3_2_4"><span class="lab"><font color="Green" title="E18">A29</font></span></a>, <a class="txt" href="#E2:3_2_4_1_1_2"><span class="lab"><font color="Green" title="E20">A33</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_2_4_1_1_2"><span class="lab"><font color="Green" title="E21">A34</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font> = 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_4"><span class="lab"><font color="Green" title="E18">A29</font></span></a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">CAT_3:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_2_4_1_1_2"><span class="lab"><font color="Green" title="E21">A34</font></span></a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">CAT_3:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c9">G</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:3_2"><span class="lab"><font color="Green" title="E7">A13</font></span></a>, <a class="txt" href="#E15:3_2"><span class="lab"><font color="Green" title="E11">A17</font></span></a>, <a class="txt" href="#E25:3_2"><span class="lab"><font color="Green" title="E17">A23</font></span></a>, <a class="ref" href="cat_3.html#T20" title="CAT_3:th.20">CAT_3:20</a></span>
<br/>.= 
<font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_2_4_1_1_2"><span class="lab"><font color="Green" title="E21">A34</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_2_4_1_1_2"><span class="lab"><font color="Green" title="E21">A34</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_4"><span class="lab"><font color="Green" title="E18">A29</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:3_2_4"/>
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E30:3_2"/><span class="lab"><font color="Green" title="E18">A35</font></span>: 
 <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T1" title="CAT_3:th.1">CAT_3:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c17">F9</font></span><br/>

<span class="kw">thus </span><a NAME="E31:3_2"/>
<font color="Maroon" title="c17">F9</font> is    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c12">c</font>,<font color="Maroon" title="c4">I</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="cat_3.html#D15" title="CAT_3:def.15">CAT_3:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" title="c1">C</font><br/> for <font color="Olive">b<sub>2</sub></font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c4">I</font>  holds <br/>(  not  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive">b<sub>2</sub></font> or  ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c12">c</font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>  holds <br/>(  not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c12">c</font> or ( (  ex <font color="Olive">b<sub>5</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>5</sub></font> ) or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> ) &amp; (  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> or for <font color="Olive">b<sub>5</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>5</sub></font> ) ) ) ) ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c18">d</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c4">I</font>  holds <br/>(  not  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive">b<sub>1</sub></font> or  ex <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#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>  holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font> or ( (  ex <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>4</sub></font> ) or <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> ) &amp; (  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> or for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>4</sub></font> ) ) ) ) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c19">F99</font> be    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c4">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c19">F99</font> or  ex <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#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</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#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font> or ( (  ex <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font> ) or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; (  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> or for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font> ) ) ) ) ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E32:3_2"/><span class="lab"><font color="Green" title="E19">A36</font></span>: 
 <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c19">F99</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</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#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font> or ( (  ex <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font> ) or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; (  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> or for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>3</sub></font> ) ) ) ) ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c20">G99</font> = <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c20">G99</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c18">d</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:3_2_5"/><span class="lab"><font color="Green" title="E20">A37</font></span>: 
<font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c20">G99</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c18">d</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">y</font></span><br/><span class="kw">then </span><font color="Maroon" title="c20">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">y</font> = 
<font color="Maroon" title="c20">G99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<font color="Maroon" title="c19">F99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_5"><span class="lab"><font color="Green" title="E20">A37</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_5"><span class="lab"><font color="Green" title="E20">A37</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c20">G99</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">y</font> = 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_5"><span class="lab"><font color="Green" title="E20">A37</font></span></a>, <a class="ref" href="cat_3.html#D3" title="CAT_3:def.3">CAT_3:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c18">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_5"><span class="lab"><font color="Green" title="E20">A37</font></span></a>, <a class="ref" href="cat_3.html#T45" title="CAT_3:th.45">CAT_3:45</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c18">d</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_5"><span class="lab"><font color="Green" title="E20">A37</font></span></a>, <a class="ref" href="cat_3.html#T2" title="CAT_3:th.2">CAT_3:2</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E35:3_2"/><span class="kw">then </span>
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c20">G99</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T1" title="CAT_3:th.1">CAT_3:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">G99</font> = <font color="Maroon" title="c20">G99</font> as    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D14" title="CAT_3:def.14">CAT_3:def 14</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c21">G99</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:3_2_6"/><span class="lab"><font color="Green" title="E20">A38</font></span>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c21">G99</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font></span><br/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A39</font></span>: <font color="Maroon" title="c9">G</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font> = 
<font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_6"><span class="lab"><font color="Green" title="E20">A38</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_6"><span class="lab"><font color="Green" title="E20">A38</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
;<br/><font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font> = 
<font color="Maroon" title="c19">F99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_6"><span class="lab"><font color="Green" title="E20">A38</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<font color="Maroon" title="c21">G99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_6"><span class="lab"><font color="Green" title="E20">A38</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c21">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_6"><span class="lab"><font color="Green" title="E20">A38</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c11">G9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font> = 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:3_2"><span class="lab"><font color="Green" title="E7">A13</font></span></a>, <a class="txt" href="#E30:3_2"><span class="lab"><font color="Green" title="E18">A35</font></span></a>, <a class="txt" href="#E32:3_2"><span class="lab"><font color="Green" title="E19">A36</font></span></a>, <a class="txt" href="#E1:3_2_6"><span class="lab"><font color="Green" title="E20">A38</font></span></a>, <a class="txt" href="#E2:3_2_6"><span class="lab"><font color="Green" title="E21">A39</font></span></a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">CAT_3:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c21">G99</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_6"><span class="lab"><font color="Green" title="E20">A38</font></span></a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">CAT_3:def 4</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E38:3_2"/><span class="kw">then </span>
 <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c11">G9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c21">G99</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T1" title="CAT_3:th.1">CAT_3:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">h9</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E40:3_2"/><span class="lab"><font color="Green" title="E20">A40</font></span>: 
<font color="Maroon" title="c22">h9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c10">a</font>
 <span class="kw">and </span><br/><a NAME="E41:3_2"/><span class="lab"><font color="Green" title="E21">A41</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c10">a</font> holds <br/>( ( for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">y</font> ) iff <font color="Maroon" title="c22">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:3_2"><span class="lab"><font color="Green" title="E8">A14</font></span></a>, <a class="ref" href="cat_3.html#D15" title="CAT_3:def.15">CAT_3:def 15</a></span>;<br/>
<a NAME="E42:3_2"/><span class="lab"><font color="Green" title="E22">A42</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">g</font> = <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font>;<br/>
<a NAME="E43:3_2"/><span class="lab"><font color="Green" title="E23">A43</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_2"><span class="lab"><font color="Green" title="E6">A12</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="cat_3.html#T45" title="CAT_3:th.45">CAT_3:45</a></span>;<br/>
<span class="lab"><font color="Green" title="E24">A44</font></span>: <font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c17">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_2"><span class="lab"><font color="Green" title="E6">A12</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:3_2"><span class="lab"><font color="Green" title="E14">A20</font></span></a>, <a class="txt" href="#E42:3_2"><span class="lab"><font color="Green" title="E22">A42</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
<font color="Maroon" title="c14">p2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:3_2"><span class="lab"><font color="Green" title="E22">A42</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/>
<span class="kw">then </span> <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c14">p2</font> = 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c17">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_2"><span class="lab"><font color="Green" title="E6">A12</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">CAT_3:def 4</a></span>
<br/>.= 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_2"><span class="lab"><font color="Green" title="E6">A12</font></span></a>, <a class="txt" href="#E32:3_2"><span class="lab"><font color="Green" title="E19">A36</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="cat_3.html#D4" title="CAT_3:def.4">CAT_3:def 4</a></span>
;<br/>
<a NAME="E46:3_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c14">p2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:3_2"><span class="lab"><font color="Green" title="E23">A43</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E48:3_2"/><span class="lab"><font color="Green" title="E25">A45</font></span>: 
<font color="Maroon" title="c24">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font>
 <span class="kw">and </span><br/><a NAME="E49:3_2"/><span class="lab"><font color="Green" title="E26">A46</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font> holds <br/>( ( <font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h9</font> &amp; <font color="Maroon" title="c14">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> ) iff <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:3_2"><span class="lab"><font color="Green" title="E11">A17</font></span></a>, <a class="txt" href="#E17:3_2"><span class="lab"><font color="Green" title="E13">A19</font></span></a>, <a class="txt" href="#E40:3_2"><span class="lab"><font color="Green" title="E20">A40</font></span></a>, <a class="ref" href="cat_3.html#D16" title="CAT_3:def.16">CAT_3:def 16</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c24">h</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</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#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font> or ( (  ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font> ) or <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; (  not <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E50:3_2"/>
<font color="Maroon" title="c24">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:3_2"><span class="lab"><font color="Green" title="E25">A45</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font> or ( (  ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font> ) or <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; (  not <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>2</sub></font> ) ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c25">k</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font> or ( (  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font> ) or <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k</font> ) &amp; (  not <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k</font> or for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E51:3_2"/><span class="lab"><font color="Green" title="E27">A47</font></span>: 
<font color="Maroon" title="c25">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c12">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font> ) or <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k</font> ) &amp; (  not <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k</font> or for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E52:3_2"/>
( ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y</font> ) implies <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k</font> or for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_9"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:3_2_9"/><span class="lab"><font color="Green" title="E28">A48</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:3_2"><span class="lab"><font color="Green" title="E27">A47</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<a NAME="E2:3_2_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A49</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:3_2"><span class="lab"><font color="Green" title="E9">A15</font></span></a>, <a class="txt" href="#E15:3_2"><span class="lab"><font color="Green" title="E11">A17</font></span></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:3_2_9"/><span class="lab"><font color="Green" title="E30">A50</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k</font></span><br/>

<a NAME="E4:3_2_9"/><span class="lab"><font color="Green" title="E31">A51</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:3_2_9_1"/><span class="lab"><font color="Green" title="E32">A52</font></span>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span><br/>

<a NAME="E2:3_2_9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A53</font></span>: 
not <font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="lab"><font color="Green" title="E34">A54</font></span>: <font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font> = 
<font color="Maroon" title="c17">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_9_1"><span class="lab"><font color="Green" title="E32">A52</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:3_2"><span class="lab"><font color="Green" title="E14">A20</font></span></a>, <a class="txt" href="#E22:3_2"><span class="lab"><font color="Green" title="E16">A22</font></span></a>, <a class="txt" href="#E1:3_2_9_1"><span class="lab"><font color="Green" title="E32">A52</font></span></a>, <a class="txt" href="#E2:3_2_9_1"><span class="lab"><font color="Green" title="E33">A53</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_9_1"><span class="lab"><font color="Green" title="E32">A52</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
;<br/>
<a NAME="E4:3_2_9_1"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_9_1"><span class="lab"><font color="Green" title="E32">A52</font></span></a>, <a class="ref" href="cat_3.html#T45" title="CAT_3:th.45">CAT_3:45</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:3_2"><span class="lab"><font color="Green" title="E9">A15</font></span></a>, <a class="txt" href="#E15:3_2"><span class="lab"><font color="Green" title="E11">A17</font></span></a>, <a class="txt" href="#E1:3_2_9"><span class="lab"><font color="Green" title="E28">A48</font></span></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_9_1"><span class="lab"><font color="Green" title="E32">A52</font></span></a>, <a class="ref" href="cat_3.html#D7" title="CAT_3:def.7">CAT_3:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_2_9"><span class="lab"><font color="Green" title="E30">A50</font></span></a>, <a class="txt" href="#E1:3_2_9_1"><span class="lab"><font color="Green" title="E32">A52</font></span></a>, <a class="txt" href="#E3:3_2_9_1"><span class="lab"><font color="Green" title="E34">A54</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c19">F99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_9_1"><span class="lab"><font color="Green" title="E32">A52</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<font color="Maroon" title="c21">G99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_9_1"><span class="lab"><font color="Green" title="E32">A52</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c21">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_2_9_1"><span class="lab"><font color="Green" title="E32">A52</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:3_2_9"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:3_2"><span class="lab"><font color="Green" title="E27">A47</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<a NAME="E6:3_2_9"/><span class="kw">then </span>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:3_2"><span class="lab"><font color="Green" title="E9">A15</font></span></a>, <a class="txt" href="#E1:3_2_9"><span class="lab"><font color="Green" title="E28">A48</font></span></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></span>;<br/>
<a NAME="E7:3_2_9"/><span class="kw">then </span>
<font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c18">d</font>,<font color="Maroon" title="c10">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_2_9"><span class="lab"><font color="Green" title="E29">A49</font></span></a></span>;<br/>
<a NAME="E8:3_2_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A55</font></span>: 
<font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:3_2"><span class="lab"><font color="Green" title="E21">A41</font></span></a>, <a class="txt" href="#E4:3_2_9"><span class="lab"><font color="Green" title="E31">A51</font></span></a></span>;<br/>
<a NAME="E9:3_2_9"/>
<font color="Maroon" title="c14">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_2"><span class="lab"><font color="Green" title="E6">A12</font></span></a>, <a class="txt" href="#E44:3_2"><span class="lab"><font color="Green" title="E24">A44</font></span></a>, <a class="txt" href="#E3:3_2_9"><span class="lab"><font color="Green" title="E30">A50</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:3_2_9"/>
<font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:3_2"><span class="lab"><font color="Green" title="E26">A46</font></span></a>, <a class="txt" href="#E51:3_2"><span class="lab"><font color="Green" title="E27">A47</font></span></a>, <a class="txt" href="#E8:3_2_9"><span class="lab"><font color="Green" title="E32">A55</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E53:3_2"/><span class="lab"><font color="Green" title="E28">A56</font></span>: 
<font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c26">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E54:3_2"/><span class="lab"><font color="Green" title="E29">A57</font></span>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_10"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_10_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:3_2_10_1"/>
( <font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> or <font color="Maroon" title="c26">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_10_1_1"><span class="kw">suppose </span></a><a NAME="E1:3_2_10_1_1"/><span class="lab"><font color="Green" title="E30">A58</font></span>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span><br/><div class="add"><a NAME="E2:3_2_10_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A59</font></span>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font> = 
<font color="Maroon" title="c17">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:3_2"><span class="lab"><font color="Green" title="E29">A57</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="cat_3.html#K1" title="CAT_3:func.1">.--&gt;</a> <font color="Maroon" title="c14">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:3_2"><span class="lab"><font color="Green" title="E14">A20</font></span></a>, <a class="txt" href="#E2:3_2_10_1_1"><span class="lab"><font color="Green" title="E31">A59</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
<font color="Maroon" title="c14">p2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_2_10_1_1"><span class="lab"><font color="Green" title="E31">A59</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:3_2_10_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:3_2"><span class="lab"><font color="Green" title="E26">A46</font></span></a>, <a class="txt" href="#E51:3_2"><span class="lab"><font color="Green" title="E27">A47</font></span></a>, <a class="txt" href="#E53:3_2"><span class="lab"><font color="Green" title="E28">A56</font></span></a>, <a class="txt" href="#E1:3_2_10_1_1"><span class="lab"><font color="Green" title="E30">A58</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_2_10_1_2"><span class="kw">suppose </span></a><a NAME="E1:3_2_10_1_2"/><span class="lab"><font color="Green" title="E30">A60</font></span>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span><br/><div class="add"><a NAME="E2:3_2_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A61</font></span>: 
not <font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:3_2_10_1_2"/><span class="lab"><font color="Green" title="E32">A62</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:3_2"><span class="lab"><font color="Green" title="E27">A47</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/><a NAME="E4:3_2_10_1_2"/><span class="lab"><font color="Green" title="E33">A63</font></span>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:3_2"><span class="lab"><font color="Green" title="E29">A57</font></span></a>, <a class="txt" href="#E1:3_2_10_1_2"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/><a NAME="E5:3_2_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A64</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T45" title="CAT_3:th.45">CAT_3:45</a></span>;<br/><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font> = 
<font color="Maroon" title="c17">F9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:3_2"><span class="lab"><font color="Green" title="E29">A57</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:3_2"><span class="lab"><font color="Green" title="E14">A20</font></span></a>, <a class="txt" href="#E22:3_2"><span class="lab"><font color="Green" title="E16">A22</font></span></a>, <a class="txt" href="#E54:3_2"><span class="lab"><font color="Green" title="E29">A57</font></span></a>, <a class="txt" href="#E2:3_2_10_1_2"><span class="lab"><font color="Green" title="E31">A61</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_2_10_1_2"><span class="lab"><font color="Green" title="E33">A63</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c13">p1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_2_10_1_2"><span class="lab"><font color="Green" title="E33">A63</font></span></a>, <a class="ref" href="cat_3.html#D7" title="CAT_3:def.7">CAT_3:def 7</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:3_2"><span class="lab"><font color="Green" title="E9">A15</font></span></a>, <a class="txt" href="#E15:3_2"><span class="lab"><font color="Green" title="E11">A17</font></span></a>, <a class="txt" href="#E3:3_2_10_1_2"><span class="lab"><font color="Green" title="E32">A62</font></span></a>, <a class="txt" href="#E5:3_2_10_1_2"><span class="lab"><font color="Green" title="E34">A64</font></span></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c22">h9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:3_2"><span class="lab"><font color="Green" title="E26">A46</font></span></a>, <a class="txt" href="#E51:3_2"><span class="lab"><font color="Green" title="E27">A47</font></span></a>, <a class="txt" href="#E53:3_2"><span class="lab"><font color="Green" title="E28">A56</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c21">G99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:3_2"><span class="lab"><font color="Green" title="E20">A40</font></span></a>, <a class="txt" href="#E41:3_2"><span class="lab"><font color="Green" title="E21">A41</font></span></a>, <a class="txt" href="#E4:3_2_10_1_2"><span class="lab"><font color="Green" title="E33">A63</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c21">G99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_2_10_1_2"><span class="lab"><font color="Green" title="E33">A63</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
<br/>.= 
<font color="Maroon" title="c19">F99</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_2_10_1_2"><span class="lab"><font color="Green" title="E33">A63</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:3_2"><span class="lab"><font color="Green" title="E29">A57</font></span></a>, <a class="ref" href="funct_2.html#D14" title="FUNCT_2:def.14">FUNCT_2:def 14</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E56:3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">F99</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c3">I</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="cat_4.html#D3" target="_self" title="CAT_4:def.3">CAT_4:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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" title="c1">C</font>  ex <font color="Olive" title="b2">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b3">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b2">a</font>,<font color="Maroon" title="c3">I</font> st <br/>(  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b3">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b3">F9</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font> st <br/>(  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b2">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Olive" title="b1">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b2">F9</font> )</span><br/>



<a NAME="E6:3"/><span class="lab"><font color="Green" title="E5">A65</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">I</font>
 
;<br/>
<a NAME="E7:3"/><span class="lab"><font color="Green" title="E6">A66</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">I</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">I</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  holds <br/> ex <font color="Olive" title="b2">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b3">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b2">a</font>,<font color="Maroon" title="c5">I</font> st <br/>(  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b3">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b3">F9</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">I</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c5">I</font> st <br/>(  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b2">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Olive" title="b1">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b2">F9</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:3_3"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c5">I</font> st <br/>(  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b2">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Olive" title="b1">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b2">F9</font> )</span><br/>

<a NAME="E2:3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A67</font></span>: 
<font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E3:3_3"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T25" title="RELSET_1:th.25">RELSET_1:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">F9</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">I</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_3"><span class="lab"><font color="Green" title="E7">A67</font></span></a>, <a class="ref" href="cat_3.html#T46" title="CAT_3:th.46">CAT_3:46</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">a</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">I</font> st <br/>(  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b1">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c2">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b1">F9</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">F9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c7">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c2">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c7">F9</font> )</span><br/>

<a NAME="E5:3_3"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font> holds <br/><span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c7">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E6:3_3"/>
 <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c7">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T1" title="CAT_3:th.1">CAT_3:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c7">F9</font></span><br/>

<span class="kw">thus </span><a NAME="E7:3_3"/>
<font color="Maroon" title="c2">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c7">F9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E2">A8</font></span></a>, <a class="txt" href="#E2:3_3"><span class="lab"><font color="Green" title="E7">A67</font></span></a>, <a class="ref" href="cat_3.html#T55" title="CAT_3:th.55">CAT_3:55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:3"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E7:3"><span class="lab"><font color="Green" title="E6">A66</font></span></a>, <a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E4">A10</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E9:3"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ex <font color="Olive" title="b2">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font> st <br/>(  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b2">F9</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Olive" title="b1">a</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b2">F9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:3"><span class="lab"><font color="Green" title="E5">A65</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
