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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="altcat_1.html#V9" title="ALTCAT_1:attr.9">semi-functional</a>   <a href="yellow18.html#V2" title="YELLOW18:attr.2">para-functional</a>  <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b3">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>  st <font color="Olive" title="b3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> holds <br/><font color="Olive" title="b3">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  for <font color="Olive" title="b1">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>  st <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> holds <br/><font color="Olive" title="b1">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:80"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>  st <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> holds <br/><font color="Olive" title="b1">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> implies <font color="Maroon" title="c4">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:80"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> or <font color="Maroon" title="c4">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:80"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> ;<br/>
<a NAME="E5:80"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="yellow18.html#T35" target="_self" onmouseover="rs('yellow18/T35')" onmouseout="rh()">Th35</a></span>;<br/>
<a NAME="E6:80"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A4</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/>
<a NAME="E7:80"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="funct_1.html#T39" onmouseover="rs('funct_1/T39')" onmouseout="rh()">FUNCT_1:39</a></span>;<br/>
<a NAME="E8:80"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="funct_1.html#T39" onmouseover="rs('funct_1/T39')" onmouseout="rh()">FUNCT_1:39</a></span>;<br/>
<a NAME="E9:80"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="yellow18.html#T35" target="_self" onmouseover="rs('yellow18/T35')" onmouseout="rh()">Th35</a></span>;<br/>
<a NAME="E10:80"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E6:80"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E8:80"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="yellow18.html#T36" target="_self" onmouseover="rs('yellow18/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<a NAME="E11:80"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E7:80"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E9:80"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="ref" href="yellow18.html#T36" target="_self" onmouseover="rs('yellow18/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<a NAME="E12:80"/><span class="lab"><font color="Green" title="E46">A10</font></span>: 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:80"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="ref" href="yellow18.html#T37" target="_self" onmouseover="rs('yellow18/T37')" onmouseout="rh()">Th37</a></span>;<br/>
<a NAME="E13:80"/>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:80"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="ref" href="yellow18.html#T37" target="_self" onmouseover="rs('yellow18/T37')" onmouseout="rh()">Th37</a></span>;<br/>
<a NAME="E14:80"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A11</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c4">f</font>
 
;<br/>
<a NAME="E15:80"/><span class="lab"><font color="Green" title="E48">A12</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:80"><span class="lab"><font color="Green" title="E46">A10</font></span></a></span>;<br/>
<a NAME="E16:80"/><span class="kw">then </span>
( <font color="Maroon" title="c4">f</font> is  <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:80"><span class="lab"><font color="Green" title="E47">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:80"/>
( <font color="Maroon" title="c4">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E14:80"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="txt" href="#E15:80"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="ref" href="altcat_3.html#D4" onmouseover="rs('altcat_3/D4')" onmouseout="rh()">ALTCAT_3:def 4</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="altcat_3.html#D5" onmouseover="rs('altcat_3/D5')" onmouseout="rh()">ALTCAT_3:def 5</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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