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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font>, <font color="Olive" title="b3">o3</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">o1</font>,<font color="Olive" title="b2">o2</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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">o2</font>,<font color="Olive" title="b3">o3</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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b3">o3</font>,<font color="Olive" title="b1">o1</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="b4">A</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font><br/> for <font color="Olive" title="b5">B</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">o2</font>,<font color="Olive" title="b3">o3</font>  st <font color="Olive" title="b4">A</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  &amp; <font color="Olive" title="b5">B</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  holds <br/><font color="Olive" title="b5">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b4">A</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">o1</font>, <font color="Maroon" title="c3">o2</font>, <font color="Maroon" title="c4">o3</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">o1</font>,<font color="Maroon" title="c3">o2</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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o3</font>,<font color="Maroon" title="c2">o1</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">A</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font><br/> for <font color="Olive" title="b2">B</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font>  st <font color="Olive" title="b1">A</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  &amp; <font color="Olive" title="b2">B</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  holds <br/><font color="Olive" title="b2">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">A</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:29"/><span class="lab"><font color="Green" title="E23">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">o1</font>,<font color="Maroon" title="c3">o2</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> 
 <span class="kw">and </span><br/><a NAME="E2:29"/><span class="lab"><font color="Green" title="E24">A2</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="c3">o2</font>,<font color="Maroon" title="c4">o3</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> 
 <span class="kw">and </span><br/><a NAME="E3:29"/><span class="lab"><font color="Green" title="E25">A3</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="c4">o3</font>,<font color="Maroon" title="c2">o1</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font><br/> for <font color="Olive" title="b2">B</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font>  st <font color="Olive" title="b1">A</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  &amp; <font color="Olive" title="b2">B</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  holds <br/><font color="Olive" title="b2">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">A</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> </span><br/>

<a NAME="E4:29"/><span class="lab"><font color="Green" title="E26">A4</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="c3">o2</font>,<font color="Maroon" title="c2">o1</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></span>;<br/>
<a NAME="E5:29"/><span class="lab"><font color="Green" title="E27">A5</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="c4">o3</font>,<font color="Maroon" title="c3">o2</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font>  st <font color="Maroon" title="c5">A</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  &amp; <font color="Olive" title="b1">B</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  holds <br/><font color="Olive" title="b1">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">A</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">B</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</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="c5">A</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  &amp; <font color="Maroon" title="c6">B</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  implies <font color="Maroon" title="c6">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">A</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:29"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c5">A</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> 
 <span class="kw">and </span><br/><a NAME="E7:29"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c6">B</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</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="c6">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">A</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">A1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c2">o1</font><span class="kw"> such that </span><br/><a NAME="E9:29"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
<font color="Maroon" title="c7">A1</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="altcat_3.html#D3" target="_self" title="ALTCAT_3:def.3">Def3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">B1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">o3</font>,<font color="Maroon" title="c3">o2</font><span class="kw"> such that </span><br/><a NAME="E11:29"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<font color="Maroon" title="c8">B1</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c6">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="altcat_3.html#D3" target="_self" title="ALTCAT_3:def.3">Def3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">G</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">o3</font>,<font color="Maroon" title="c2">o1</font><span class="kw"> such that </span><br/><a NAME="E13:29"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c8">B1</font>
 ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">G</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="altcat_3.html#D3" target="_self" title="ALTCAT_3:def.3">ALTCAT_3:def 3</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="c9">G</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c6">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">A</font></span><br/>

<a NAME="E14:29"/><span class="lab"><font color="Green" title="E33">A11</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="c3">o2</font>,<font color="Maroon" title="c3">o2</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#T23" title="ALTCAT_1:th.23">ALTCAT_1:23</a></span>;<br/>
<font color="Maroon" title="c9">G</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">A</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E13:29"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="altcat_1.html#T25" title="ALTCAT_1:th.25">ALTCAT_1:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">A1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">B1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="altcat_1.html#T25" title="ALTCAT_1:th.25">ALTCAT_1:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">A1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p2">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o2</font></span>)</span></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="altcat_3.html#D1" target="_self" title="ALTCAT_3:def.1">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c7">A1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o2</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E14:29"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="altcat_1.html#T25" title="ALTCAT_1:th.25">ALTCAT_1:25</a></span>
<br/>.= 
<font color="Maroon" title="c7">A1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="altcat_1.html#T24" title="ALTCAT_1:th.24">ALTCAT_1:24</a></span>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">o1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="altcat_3.html#D1" target="_self" title="ALTCAT_3:def.1">Def1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:29"/>
<font color="Maroon" title="c9">G</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c6">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_3.html#D1" target="_self" title="ALTCAT_3:def.1">Def1</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/>


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