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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="altcat_1.html#L1">AltGraph</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#V1">reflexive</a>  <a href="altcat_1.html#L1">AltGraph</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="functor0.html#V8">feasible</a>  <a href="functor0.html#L2">FunctorStr</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="functor0.html#L2">FunctorStr</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><a NAME="E1:11"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="functor0.html#V17">faithful</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="functor0.html#V17">faithful</a> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K13">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">c<sub>4</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E4:11"/><i><font color="Green">E7</font></i>: 
( the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>4</sub></font> is <a href="msualg_3.html#V1">"1-1"</a> &amp; the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>5</sub></font> is <a href="msualg_3.html#V1">"1-1"</a> )
 
<b>by </b><i><a class="txt" href="functor1.html#E1:11"><i><font color="Green">E6</font></i></a>, <a class="ref" href="functor0.html#D31">FUNCTOR0:def 31</a></i>;<br/>
<a NAME="E5:11"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="altcat_2.html#K3">**</a> the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="functor0.html#D37">FUNCTOR0:def 37</a></i>;<br/>
<a NAME="E6:11"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:11_1"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E3:11_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="altcat_2.html#K3">**</a> the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="functor1.html#E1:11_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E11</font></i>: <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor1.html#E5:11"><i><font color="Green">E8</font></i></a>, <a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor1.html#E1:11_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>
<a NAME="E5:11_1"/><i><font color="Green">E12</font></i>: 
the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="functor1.html#E4:11"><i><font color="Green">E7</font></i></a>, <a class="txt" href="functor1.html#E2:11_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>;<br/>
<a NAME="E6:11_1"/>
the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="functor1.html#E4:11"><i><font color="Green">E7</font></i></a>, <a class="txt" href="functor1.html#E1:11_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>;<br/>
<b>hence </b><a NAME="E7:11_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="functor1.html#E4:11_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="functor1.html#E5:11_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_1.html#T46">FUNCT_1:46</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:11"/>
the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K13">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="msualg_3.html#V1">"1-1"</a>
 <b>by </b><i><a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D31">FUNCTOR0:def 31</a><br/>


</div>
