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<b>let </b><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being <a href="functor0.html#V10" title="FUNCTOR0:attr.10">Covariant</a>  <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> holds <br/> ( <font color="Olive" title="b1">F</font> is <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a> iff for <font color="Olive" title="b2">o1</font>, <font color="Olive" title="b3">o2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C1</font> holds   <a href="functor0.html#K5" title="FUNCTOR0:func.5">Morph-Map</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">o1</font>,<font color="Olive" title="b3">o2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )</span><br/><b>let </b><font color="Maroon" title="c3">F</font> be  <a href="functor0.html#V10" title="FUNCTOR0:attr.10">Covariant</a>  <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a> iff for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C1</font> holds   <a href="functor0.html#K5" title="FUNCTOR0:func.5">Morph-Map</a> <font color="Maroon" title="c3">F</font>,<font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )</span><br/>



<b>set </b><font color="Maroon" title="c4">I</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C1</font> holds   <a href="functor0.html#K5" title="FUNCTOR0:func.5">Morph-Map</a> <font color="Maroon" title="c3">F</font>,<font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> ) implies <font color="Maroon" title="c3">F</font> is <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C1</font> holds   <a href="functor0.html#K5" title="FUNCTOR0:func.5">Morph-Map</a> <font color="Maroon" title="c3">F</font>,<font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a></span><br/><b>let </b><font color="Maroon" title="c5">o1</font>, <font color="Maroon" title="c6">o2</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="functor0.html#K5" title="FUNCTOR0:func.5">Morph-Map</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c5">o1</font>,<font color="Maroon" title="c6">o2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a></span><br/><a NAME="E2:20_1"/><i><font color="Green" title="E12">A2</font></i>: 
the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c3">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>
 <b>by </b><i><a class="txt" href="#E1:20_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="functor0.html#D31" title="FUNCTOR0:def.31">FUNCTOR0:def 31</a></i>;<br/><a NAME="E3:20_1"/><i><font color="Green" title="E13">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o1</font>,<font color="Maroon" title="c6">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:20_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E5:20_1"/>
 <a href="functor0.html#K5" title="FUNCTOR0:func.5">Morph-Map</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c5">o1</font>,<font color="Maroon" title="c6">o2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="#E2:20_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E3:20_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="msualg_3.html#D2" title="MSUALG_3:def.2">MSUALG_3:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:20"/><i><font color="Green" title="E11">A4</font></i>: 
for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C1</font> holds   <a href="functor0.html#K5" title="FUNCTOR0:func.5">Morph-Map</a> <font color="Maroon" title="c3">F</font>,<font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">F</font> is <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a></span><br/>

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D2" title="MSUALG_3:def.2">MSUALG_3:def 2</a>,<a class="ref" href="functor0.html#D31" title="FUNCTOR0:def.31">FUNCTOR0:def 31</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c3">F</font> or  not the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )</span><br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c3">F</font> or  not the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> or <font color="Maroon" title="c6">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )</span><br/>



<b>assume </b><a NAME="E3:20"/><i><font color="Green" title="E12">A5</font></i>: 
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c3">F</font> &amp; the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a></span><br/>

<a NAME="E4:20"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">o1</font>, <font color="Maroon" title="c8">o2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:20"/><i><font color="Green" title="E13">A6</font></i>: 
( <font color="Maroon" title="c7">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font> &amp; <font color="Maroon" title="c8">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font> &amp; <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="#E3:20"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">o1</font> = <font color="Maroon" title="c7">o1</font>, <font color="Maroon" title="c10">o2</font> = <font color="Maroon" title="c8">o2</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="txt" href="#E6:20"><i><font color="Green" title="E13">A6</font></i></a></i>;<br/>
<a NAME="E8:20"/>
the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">o1</font>,<font color="Maroon" title="c10">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K5" title="FUNCTOR0:func.5">Morph-Map</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c9">o1</font>,<font color="Maroon" title="c10">o2</font>
 
;<br/>
<b>hence </b><a NAME="E9:20"/>
<font color="Maroon" title="c6">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="#E2:20"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E3:20"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="#E6:20"><i><font color="Green" title="E13">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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