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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">C</font><br/> 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">C</font><br/> for <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Olive" title="b1">D</font><br/> for <font color="Olive" title="b6">m</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">o1</font>,<font color="Olive" title="b3">o2</font><br/> for <font color="Olive" title="b7">n</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</font>  st <font color="Olive" title="b4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o1</font> &amp; <font color="Olive" title="b5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o2</font> &amp; <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">n</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b4">p1</font>,<font color="Olive" title="b5">p2</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="b5">p2</font>,<font color="Olive" title="b4">p1</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/>( ( <font color="Olive" title="b7">n</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> implies <font color="Olive" title="b6">m</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> ) &amp; ( <font color="Olive" title="b7">n</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> implies <font color="Olive" title="b6">m</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> ) &amp; ( <font color="Olive" title="b7">n</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> implies <font color="Olive" title="b6">m</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) )</span><br/><b>let </b><font color="Maroon" title="c2">D</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">C</font>; <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">C</font><br/> for <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">p2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">D</font><br/> for <font color="Olive" title="b5">m</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="b6">n</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font>  st <font color="Olive" title="b3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o1</font> &amp; <font color="Olive" title="b4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o2</font> &amp; <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">n</font> &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">p1</font>,<font color="Olive" title="b4">p2</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="b4">p2</font>,<font color="Olive" title="b3">p1</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/>( ( <font color="Olive" title="b6">n</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> implies <font color="Olive" title="b5">m</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> ) &amp; ( <font color="Olive" title="b6">n</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> implies <font color="Olive" title="b5">m</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> ) &amp; ( <font color="Olive" title="b6">n</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> implies <font color="Olive" title="b5">m</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) )</span><br/><b>let </b><font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c4">o2</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">D</font><br/> for <font color="Olive" title="b3">m</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font><br/> for <font color="Olive" title="b4">n</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>  st <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o1</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font> &amp; <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">n</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</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">p2</font>,<font color="Olive" title="b1">p1</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/>( ( <font color="Olive" title="b4">n</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> implies <font color="Olive" title="b3">m</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> ) &amp; ( <font color="Olive" title="b4">n</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> implies <font color="Olive" title="b3">m</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> ) &amp; ( <font color="Olive" title="b4">n</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> implies <font color="Olive" title="b3">m</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) )</span><br/><b>let </b><font color="Maroon" title="c5">p1</font>, <font color="Maroon" title="c6">p2</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">D</font>; <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">m</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font><br/> for <font color="Olive" title="b2">n</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>  st <font color="Maroon" title="c5">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o1</font> &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</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="c6">p2</font>,<font color="Maroon" title="c5">p1</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/>( ( <font color="Olive" title="b2">n</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> implies <font color="Olive" title="b1">m</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> ) &amp; ( <font color="Olive" title="b2">n</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> implies <font color="Olive" title="b1">m</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> ) &amp; ( <font color="Olive" title="b2">n</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> implies <font color="Olive" title="b1">m</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) )</span><br/><b>let </b><font color="Maroon" title="c7">m</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>; <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">n</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>  st <font color="Maroon" title="c5">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o1</font> &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font> &amp; <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</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="c6">p2</font>,<font color="Maroon" title="c5">p1</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/>( ( <font color="Olive" title="b1">n</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> implies <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> ) &amp; ( <font color="Olive" title="b1">n</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> implies <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> ) &amp; ( <font color="Olive" title="b1">n</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> implies <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) )</span><br/><b>let </b><font color="Maroon" title="c8">n</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o1</font> &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font> &amp; <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</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="c6">p2</font>,<font color="Maroon" title="c5">p1</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 ( ( <font color="Maroon" title="c8">n</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> implies <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> ) &amp; ( <font color="Maroon" title="c8">n</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> implies <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> ) &amp; ( <font color="Maroon" title="c8">n</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> implies <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) ) )</span><br/>











<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c5">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o1</font> &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font> &amp; <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</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="c6">p2</font>,<font color="Maroon" title="c5">p1</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c8">n</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> implies <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> ) &amp; ( <font color="Maroon" title="c8">n</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> implies <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> ) &amp; ( <font color="Maroon" title="c8">n</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> implies <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) )</span><br/>

<b>thus </b><a NAME="E2:45"/><i><font color="Green" title="E30">A2</font></i>: 
( <font color="Maroon" title="c8">n</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> implies <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c8">n</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> implies <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> ) &amp; ( <font color="Maroon" title="c8">n</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> implies <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon" title="c8">n</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a></span><br/>

<b>then consider </b><font color="Maroon" title="c9">B</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c5">p1</font><b> such that </b><br/><a NAME="E3:45_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c9">B</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="ref" href="altcat_3.html#D2" title="ALTCAT_3:def.2">ALTCAT_3:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">B1</font> = <font color="Maroon" title="c9">B</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c3">o1</font> <b>by </b><i><a class="txt" href="altcat_4.html#E1:45"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="altcat_2.html#T34" title="ALTCAT_2:th.34">ALTCAT_2:34</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">B1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D2" title="ALTCAT_3:def.2">ALTCAT_3:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">m</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c10">B1</font></span><br/>

<b>thus </b><a NAME="E5:45_1"/>
<font color="Maroon" title="c10">B1</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E1:45"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="altcat_4.html#E3:45_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="altcat_4.html#T38" target="_self" title="ALTCAT_4:th.38">Th38</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>thus </b><a NAME="E3:45"/><i><font color="Green" title="E31">A4</font></i>: 
( <font color="Maroon" title="c8">n</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> implies <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">n</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> implies <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_2"/>
<font color="Maroon" title="c8">n</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a></span><br/>

<b>then consider </b><font color="Maroon" title="c9">B</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c5">p1</font><b> such that </b><br/><a NAME="E3:45_2"/><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c9">B</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="ref" href="altcat_3.html#D3" title="ALTCAT_3:def.3">ALTCAT_3:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">B1</font> = <font color="Maroon" title="c9">B</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c3">o1</font> <b>by </b><i><a class="txt" href="altcat_4.html#E1:45"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="altcat_2.html#T34" title="ALTCAT_2:th.34">ALTCAT_2:34</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">B1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D3" title="ALTCAT_3:def.3">ALTCAT_3:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">B1</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c7">m</font></span><br/>

<b>thus </b><a NAME="E5:45_2"/>
<font color="Maroon" title="c10">B1</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E1:45"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="altcat_4.html#E3:45_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="altcat_4.html#T38" target="_self" title="ALTCAT_4:th.38">Th38</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="E4:45"/><i><font color="Green" title="E32">A6</font></i>: 
<font color="Maroon" title="c8">n</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a></span><br/>

<a NAME="E5:45"/>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">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="c4">o2</font>,<font color="Maroon" title="c3">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>  )
 
<b>by </b><i><a class="txt" href="altcat_4.html#E1:45"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E6:45"/>
<font color="Maroon" title="c7">m</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="txt" href="altcat_4.html#E2:45"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="altcat_4.html#E3:45"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="altcat_4.html#E4:45"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="altcat_3.html#T5" title="ALTCAT_3:th.5">ALTCAT_3:5</a>, <a class="ref" href="altcat_3.html#T6" title="ALTCAT_3:th.6">ALTCAT_3:6</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>
