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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="functor0.html#V11">Contravariant</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>set </b><font color="Maroon">c<sub>4</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/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V17">faithful</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:18_1"/><i><font color="Green">E9</font></i>: 
the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>3</sub></font> is <a href="msualg_3.html#V1">"1-1"</a>
 <b>by </b><i><a class="txt" href="altcat_4.html#E1:18_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="functor0.html#D31">FUNCTOR0:def 31</a></i>;<br/><a NAME="E3:18_1"/><i><font color="Green">E10</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <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>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> as   <a href="funct_1.html#NM1">Function</a> ;<br/><a NAME="E5:18_1"/>
 <a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</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>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E6:18_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="altcat_4.html#E2:18_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="altcat_4.html#E3:18_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msualg_3.html#D2">MSUALG_3:def 2</a></i>;<br/><b>hence </b><a NAME="E7:18_1"/>
 <a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:18"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D31">FUNCTOR0:def 31</a>,<a class="ref" href="msualg_3.html#D2">MSUALG_3:def 2</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E3:18"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E4:18"/>
 <a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:18"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="altcat_4.html#E3:18"><i><font color="Green">E9</font></i></a>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="altcat_4.html#E6:18"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E8:18"/>
the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
;<br/>
<b>hence </b><a NAME="E9:18"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="altcat_4.html#E2:18"><i><font color="Green">E8</font></i></a>, <a class="txt" href="altcat_4.html#E3:18"><i><font color="Green">E9</font></i></a>, <a class="txt" href="altcat_4.html#E6:18"><i><font color="Green">E10</font></i></a></i>;<br/>


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