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

<span class="kw">let </span><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c2">G2</font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c2">G2</font> implies <font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:41"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c2">G2</font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c2">G2</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">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span><br/>

<span class="lab"><font color="Green" title="E20">A2</font></span>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</a></span>
;<br/>
<span class="lab"><font color="Green" title="E21">A3</font></span>: the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> = 
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">G2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</a></span>
<br/>.= 
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</a></span>
;<br/>
<span class="lab"><font color="Green" title="E22">A4</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> = 
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</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="E5:41"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> holds <br/>the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">v</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="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> implies the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_4"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span><br/>

<a NAME="E2:41_4"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_4"><span class="lab"><font color="Green" title="E24">A6</font></span></a></span>;<br/>
<a NAME="E3:41_4"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:41_4"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E27">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:41_4_1"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span><br/><span class="kw">thus </span>the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> = 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:41_4_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</a></span>
<br/>.= 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:41_4_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</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><span class="lab"><font color="Green" title="E28">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:41_4_2"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span><br/><span class="kw">thus </span>the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> = 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c2">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:41_4_2"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</a></span>
<br/>.= 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:41_4_2"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</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">thus </span><a NAME="E6:41_4"/>
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_4"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E4:41_4"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E5:41_4"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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="E6:41"/><span class="lab"><font color="Green" title="E24">A13</font></span>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/>
<span class="lab"><font color="Green" title="E25">A14</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> = 
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</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="E8:41"/><span class="lab"><font color="Green" title="E26">A15</font></span>: 
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> holds <br/>the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">v</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="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> implies the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_6"/><span class="lab"><font color="Green" title="E27">A16</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span><br/>

<a NAME="E2:41_6"/><span class="lab"><font color="Green" title="E28">A17</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_6"><span class="lab"><font color="Green" title="E27">A16</font></span></a></span>;<br/>
<a NAME="E3:41_6"/><span class="lab"><font color="Green" title="E29">A18</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:41_6"><span class="lab"><font color="Green" title="E28">A17</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E30">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:41_6_1"/><span class="lab"><font color="Green" title="E31">A20</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span><br/><span class="kw">thus </span>the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> = 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:41_6_1"><span class="lab"><font color="Green" title="E31">A20</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</a></span>
<br/>.= 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:41_6_1"><span class="lab"><font color="Green" title="E31">A20</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</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><span class="lab"><font color="Green" title="E31">A21</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_6_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:41_6_2"/><span class="lab"><font color="Green" title="E32">A22</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font></span><br/><span class="kw">thus </span>the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> = 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c2">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:41_6_2"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</a></span>
<br/>.= 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:41_6_2"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</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">thus </span><a NAME="E6:41_6"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_6"><span class="lab"><font color="Green" title="E29">A18</font></span></a>, <a class="txt" href="#E4:41_6"><span class="lab"><font color="Green" title="E30">A19</font></span></a>, <a class="txt" href="#E5:41_6"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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">thus </span><a NAME="E9:41"/>
<font color="Maroon" title="c1">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c2">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="txt" href="#E7:41"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="txt" href="#E8:41"><span class="lab"><font color="Green" title="E26">A15</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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>
