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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">m1</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">o2</font>,<font color="Olive" title="b2">o1</font><br/> for <font color="Olive" title="b8">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><br/> for <font color="Olive" title="b9">n1</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b5">p2</font>,<font color="Olive" title="b4">p1</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="b8">n</font> &amp; <font color="Olive" title="b7">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">n1</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="b6">m</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b7">m1</font> implies <font color="Olive" title="b8">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b9">n1</font> ) &amp; ( <font color="Olive" title="b8">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b9">n1</font> implies <font color="Olive" title="b6">m</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b7">m1</font> ) &amp; ( <font color="Olive" title="b6">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b7">m1</font> implies <font color="Olive" title="b8">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b9">n1</font> ) &amp; ( <font color="Olive" title="b8">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b9">n1</font> implies <font color="Olive" title="b6">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b7">m1</font> ) )</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">m1</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="b1">o1</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="b3">p1</font>,<font color="Olive" title="b4">p2</font><br/> for <font color="Olive" title="b8">n1</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b4">p2</font>,<font color="Olive" title="b3">p1</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="b7">n</font> &amp; <font color="Olive" title="b6">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">n1</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="b5">m</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b6">m1</font> implies <font color="Olive" title="b7">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b8">n1</font> ) &amp; ( <font color="Olive" title="b7">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b8">n1</font> implies <font color="Olive" title="b5">m</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b6">m1</font> ) &amp; ( <font color="Olive" title="b5">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b6">m1</font> implies <font color="Olive" title="b7">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b8">n1</font> ) &amp; ( <font color="Olive" title="b7">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b8">n1</font> implies <font color="Olive" title="b5">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b6">m1</font> ) )</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">m1</font> being  <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><br/> for <font color="Olive" title="b5">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><br/> for <font color="Olive" title="b6">n1</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">p2</font>,<font color="Olive" title="b1">p1</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="b5">n</font> &amp; <font color="Olive" title="b4">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">n1</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="b3">m</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b4">m1</font> implies <font color="Olive" title="b5">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b6">n1</font> ) &amp; ( <font color="Olive" title="b5">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b6">n1</font> implies <font color="Olive" title="b3">m</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b4">m1</font> ) &amp; ( <font color="Olive" title="b3">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b4">m1</font> implies <font color="Olive" title="b5">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b6">n1</font> ) &amp; ( <font color="Olive" title="b5">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b6">n1</font> implies <font color="Olive" title="b3">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b4">m1</font> ) )</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">m1</font> being  <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><br/> for <font color="Olive" title="b3">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><br/> for <font color="Olive" title="b4">n1</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>  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="b3">n</font> &amp; <font color="Olive" title="b2">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">n1</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">m</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b2">m1</font> implies <font color="Olive" title="b3">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b4">n1</font> ) &amp; ( <font color="Olive" title="b3">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b4">n1</font> implies <font color="Olive" title="b1">m</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b2">m1</font> ) &amp; ( <font color="Olive" title="b1">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b2">m1</font> implies <font color="Olive" title="b3">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b4">n1</font> ) &amp; ( <font color="Olive" title="b3">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b4">n1</font> implies <font color="Olive" title="b1">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b2">m1</font> ) )</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">m1</font> being  <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><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><br/> for <font color="Olive" title="b3">n1</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>  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="b2">n</font> &amp; <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n1</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="Maroon" title="c7">m</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b1">m1</font> implies <font color="Olive" title="b2">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b3">n1</font> ) &amp; ( <font color="Olive" title="b2">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b3">n1</font> implies <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="Olive" title="b1">m1</font> ) &amp; ( <font color="Maroon" title="c7">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b1">m1</font> implies <font color="Olive" title="b2">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b3">n1</font> ) &amp; ( <font color="Olive" title="b2">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b3">n1</font> implies <font color="Maroon" title="c7">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b1">m1</font> ) )</span><br/><b>let </b><font color="Maroon" title="c8">m1</font> be   <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>; <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><br/> for <font color="Olive" title="b2">n1</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>  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; <font color="Maroon" title="c8">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n1</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="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="c8">m1</font> implies <font color="Olive" title="b1">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b2">n1</font> ) &amp; ( <font color="Olive" title="b1">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b2">n1</font> implies <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="c8">m1</font> ) &amp; ( <font color="Maroon" title="c7">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c8">m1</font> implies <font color="Olive" title="b1">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b2">n1</font> ) &amp; ( <font color="Olive" title="b1">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b2">n1</font> implies <font color="Maroon" title="c7">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c8">m1</font> ) )</span><br/><b>let </b><font color="Maroon" title="c9">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"> for <font color="Olive" title="b1">n1</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>  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="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c8">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n1</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="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="c8">m1</font> implies <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b1">n1</font> ) &amp; ( <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Olive" title="b1">n1</font> implies <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="c8">m1</font> ) &amp; ( <font color="Maroon" title="c7">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c8">m1</font> implies <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b1">n1</font> ) &amp; ( <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Olive" title="b1">n1</font> implies <font color="Maroon" title="c7">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c8">m1</font> ) )</span><br/><b>let </b><font color="Maroon" title="c10">n1</font> be   <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>; <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="c9">n</font> &amp; <font color="Maroon" title="c8">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n1</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="c7">m</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c8">m1</font> implies <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c10">n1</font> ) &amp; ( <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c10">n1</font> implies <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="c8">m1</font> ) &amp; ( <font color="Maroon" title="c7">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c8">m1</font> implies <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c10">n1</font> ) &amp; ( <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c10">n1</font> implies <font color="Maroon" title="c7">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c8">m1</font> ) ) )</span><br/>















<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E28">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="c9">n</font> &amp; <font color="Maroon" title="c8">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n1</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="c7">m</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c8">m1</font> implies <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c10">n1</font> ) &amp; ( <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c10">n1</font> implies <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="c8">m1</font> ) &amp; ( <font color="Maroon" title="c7">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c8">m1</font> implies <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c10">n1</font> ) &amp; ( <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c10">n1</font> implies <font color="Maroon" title="c7">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c8">m1</font> ) )</span><br/>

<b>thus </b><a NAME="E2:43"/>
( <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="c8">m1</font> iff <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c10">n1</font> )
  <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#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c8">m1</font> iff <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c10">n1</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:43_1"/>
( <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="c8">m1</font> implies <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c10">n1</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c10">n1</font> implies <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="c8">m1</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1"/><i><font color="Green" title="E29">A2</font></i>: 
<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="c8">m1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c10">n1</font></span><br/>

<b>thus </b><font color="Maroon" title="c9">n</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c10">n1</font> = 
<font color="Maroon" title="c7">m</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c8">m1</font>
<b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="altcat_2.html#T33" title="ALTCAT_2:th.33">ALTCAT_2:33</a></i>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">o2</font>
<b>by </b><i><a class="txt" href="#E1:43_1_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</a></i>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c6">p2</font>
<b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="altcat_2.html#T35" title="ALTCAT_2:th.35">ALTCAT_2:35</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</a> <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="E2:43_1"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c10">n1</font>
 ; <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="c8">m1</font></span><br/>

<b>thus </b><font color="Maroon" title="c7">m</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c8">m1</font> = 
<font color="Maroon" title="c9">n</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c10">n1</font>
<b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="altcat_2.html#T33" title="ALTCAT_2:th.33">ALTCAT_2:33</a></i>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c6">p2</font>
<b>by </b><i><a class="txt" href="#E2:43_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</a></i>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">o2</font>
<b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="altcat_2.html#T35" title="ALTCAT_2:th.35">ALTCAT_2:35</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</a> <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="E3:43"/>
( <font color="Maroon" title="c7">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c8">m1</font> implies <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c10">n1</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c10">n1</font> implies <font color="Maroon" title="c7">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c8">m1</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_2"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c8">m1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c10">n1</font></span><br/>

<b>thus </b><font color="Maroon" title="c10">n1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c9">n</font> = 
<font color="Maroon" title="c8">m1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">m</font>
<b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="altcat_2.html#T33" title="ALTCAT_2:th.33">ALTCAT_2:33</a></i>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o1</font>
<b>by </b><i><a class="txt" href="#E1:43_2"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</a></i>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">p1</font>
<b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="altcat_2.html#T35" title="ALTCAT_2:th.35">ALTCAT_2:35</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</a> <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="E4:43"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c10">n1</font>
 ; <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#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c8">m1</font></span><br/>

<b>thus </b><font color="Maroon" title="c8">m1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">m</font> = 
<font color="Maroon" title="c10">n1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c9">n</font>
<b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="altcat_2.html#T33" title="ALTCAT_2:th.33">ALTCAT_2:33</a></i>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">p1</font>
<b>by </b><i><a class="txt" href="#E4:43"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</a></i>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o1</font>
<b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="altcat_2.html#T35" title="ALTCAT_2:th.35">ALTCAT_2:35</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
