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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="cat_6.html#NM2" title="CAT_6:NM.2">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">c1</font>, <font color="Olive" title="b2">c2</font>, <font color="Olive" title="b3">c3</font>, <font color="Olive" title="b4">c4</font>, <font color="Olive" title="b5">c5</font>, <font color="Olive" title="b6">c6</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b7">f1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">c1</font>,<font color="Olive" title="b2">c2</font><br/>  for <font color="Olive" title="b8">f2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b2">c2</font>,<font color="Olive" title="b3">c3</font><br/>  for <font color="Olive" title="b9">f3</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">c1</font>,<font color="Olive" title="b4">c4</font><br/>  for <font color="Olive" title="b10">f4</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b2">c2</font>,<font color="Olive" title="b5">c5</font><br/>  for <font color="Olive" title="b11">f5</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b3">c3</font>,<font color="Olive" title="b6">c6</font><br/>  for <font color="Olive" title="b12">f6</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b4">c4</font>,<font color="Olive" title="b5">c5</font><br/>  for <font color="Olive" title="b13">f7</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b5">c5</font>,<font color="Olive" title="b6">c6</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">c1</font>,<font color="Olive" title="b2">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b2">c2</font>,<font color="Olive" title="b3">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">c1</font>,<font color="Olive" title="b4">c4</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b2">c2</font>,<font color="Olive" title="b5">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b3">c3</font>,<font color="Olive" title="b6">c6</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b4">c4</font>,<font color="Olive" title="b5">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b5">c5</font>,<font color="Olive" title="b6">c6</font>) <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; <font color="Olive" title="b2">c2</font>,<font color="Olive" title="b8">f2</font>,<font color="Olive" title="b10">f4</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b11">f5</font>,<font color="Olive" title="b13">f7</font> holds <br/>( <font color="Olive" title="b1">c1</font>,<font color="Olive" title="b7">f1</font>,<font color="Olive" title="b9">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b10">f4</font>,<font color="Olive" title="b12">f6</font> iff ( <font color="Olive" title="b1">c1</font>,<font color="Olive" title="b8">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b7">f1</font>,<font color="Olive" title="b9">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b11">f5</font>,<font color="Olive" title="b13">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b12">f6</font> &amp; <font color="Olive" title="b10">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b12">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b9">f3</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">c1</font>, <font color="Maroon" title="c3">c2</font>, <font color="Maroon" title="c4">c3</font>, <font color="Maroon" title="c5">c4</font>, <font color="Maroon" title="c6">c5</font>, <font color="Maroon" title="c7">c6</font> be   <a href="cat_6.html#NM3" title="CAT_6:NM.3">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">  for <font color="Olive" title="b1">f1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font><br/>  for <font color="Olive" title="b2">f2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c4">c3</font><br/>  for <font color="Olive" title="b3">f3</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">c4</font><br/>  for <font color="Olive" title="b4">f4</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c6">c5</font><br/>  for <font color="Olive" title="b5">f5</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font><br/>  for <font color="Olive" title="b6">f6</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font><br/>  for <font color="Olive" title="b7">f7</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">c4</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>) <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; <font color="Maroon" title="c3">c2</font>,<font color="Olive" title="b2">f2</font>,<font color="Olive" title="b4">f4</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b5">f5</font>,<font color="Olive" title="b7">f7</font> holds <br/>( <font color="Maroon" title="c2">c1</font>,<font color="Olive" title="b1">f1</font>,<font color="Olive" title="b3">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b4">f4</font>,<font color="Olive" title="b6">f6</font> iff ( <font color="Maroon" title="c2">c1</font>,<font color="Olive" title="b2">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b3">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b5">f5</font>,<font color="Olive" title="b7">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b6">f6</font> &amp; <font color="Olive" title="b4">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">f3</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">f1</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</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">  for <font color="Olive" title="b1">f2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c4">c3</font><br/>  for <font color="Olive" title="b2">f3</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">c4</font><br/>  for <font color="Olive" title="b3">f4</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c6">c5</font><br/>  for <font color="Olive" title="b4">f5</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font><br/>  for <font color="Olive" title="b5">f6</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font><br/>  for <font color="Olive" title="b6">f7</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">c4</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>) <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; <font color="Maroon" title="c3">c2</font>,<font color="Olive" title="b1">f2</font>,<font color="Olive" title="b3">f4</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b4">f5</font>,<font color="Olive" title="b6">f7</font> holds <br/>( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c8">f1</font>,<font color="Olive" title="b2">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b3">f4</font>,<font color="Olive" title="b5">f6</font> iff ( <font color="Maroon" title="c2">c1</font>,<font color="Olive" title="b1">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font>,<font color="Olive" title="b2">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b4">f5</font>,<font color="Olive" title="b6">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b5">f6</font> &amp; <font color="Olive" title="b3">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">f3</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">f2</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c4">c3</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">  for <font color="Olive" title="b1">f3</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">c4</font><br/>  for <font color="Olive" title="b2">f4</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c6">c5</font><br/>  for <font color="Olive" title="b3">f5</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font><br/>  for <font color="Olive" title="b4">f6</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font><br/>  for <font color="Olive" title="b5">f7</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">c4</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>) <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; <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c9">f2</font>,<font color="Olive" title="b2">f4</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b3">f5</font>,<font color="Olive" title="b5">f7</font> holds <br/>( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c8">f1</font>,<font color="Olive" title="b1">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b2">f4</font>,<font color="Olive" title="b4">f6</font> iff ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font>,<font color="Olive" title="b1">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b3">f5</font>,<font color="Olive" title="b5">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">f6</font> &amp; <font color="Olive" title="b2">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">f3</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c10">f3</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">c4</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">  for <font color="Olive" title="b1">f4</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c6">c5</font><br/>  for <font color="Olive" title="b2">f5</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font><br/>  for <font color="Olive" title="b3">f6</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font><br/>  for <font color="Olive" title="b4">f7</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">c4</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>) <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; <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c9">f2</font>,<font color="Olive" title="b1">f4</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b2">f5</font>,<font color="Olive" title="b4">f7</font> holds <br/>( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b1">f4</font>,<font color="Olive" title="b3">f6</font> iff ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b2">f5</font>,<font color="Olive" title="b4">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">f6</font> &amp; <font color="Olive" title="b1">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c11">f4</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c6">c5</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">  for <font color="Olive" title="b1">f5</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font><br/>  for <font color="Olive" title="b2">f6</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font><br/>  for <font color="Olive" title="b3">f7</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">c4</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>) <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; <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c9">f2</font>,<font color="Maroon" title="c11">f4</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b1">f5</font>,<font color="Olive" title="b3">f7</font> holds <br/>( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c11">f4</font>,<font color="Olive" title="b2">f6</font> iff ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b1">f5</font>,<font color="Olive" title="b3">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">f6</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">f5</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</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">  for <font color="Olive" title="b1">f6</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font><br/>  for <font color="Olive" title="b2">f7</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">c4</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>) <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; <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c9">f2</font>,<font color="Maroon" title="c11">f4</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c12">f5</font>,<font color="Olive" title="b2">f7</font> holds <br/>( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c11">f4</font>,<font color="Olive" title="b1">f6</font> iff ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c12">f5</font>,<font color="Olive" title="b2">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">f6</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c13">f6</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</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">  for <font color="Olive" title="b1">f7</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">c4</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>) <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; <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c9">f2</font>,<font color="Maroon" title="c11">f4</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c12">f5</font>,<font color="Olive" title="b1">f7</font> holds <br/>( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c11">f4</font>,<font color="Maroon" title="c13">f6</font> iff ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c12">f5</font>,<font color="Olive" title="b1">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c14">f7</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</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"> (  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">c4</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>) <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; <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c9">f2</font>,<font color="Maroon" title="c11">f4</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c12">f5</font>,<font color="Maroon" title="c14">f7</font> implies ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c11">f4</font>,<font color="Maroon" title="c13">f6</font> iff ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c12">f5</font>,<font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font> ) ) )</span><br/>

















<span class="kw">assume </span><a NAME="E1:75"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">c4</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c6">c5</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c6">c5</font>,<font color="Maroon" title="c7">c6</font>) <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"> (  not <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c9">f2</font>,<font color="Maroon" title="c11">f4</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c12">f5</font>,<font color="Maroon" title="c14">f7</font> or ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c11">f4</font>,<font color="Maroon" title="c13">f6</font> iff ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c12">f5</font>,<font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:75"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c9">f2</font>,<font color="Maroon" title="c11">f4</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c12">f5</font>,<font color="Maroon" title="c14">f7</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="c2">c1</font>,<font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c11">f4</font>,<font color="Maroon" title="c13">f6</font> iff ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c12">f5</font>,<font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font> ) )</span><br/>

<a NAME="E3:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A3</font></span>: 
( <font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">f4</font> &amp; (  for <font color="Olive" title="b1">d1</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b2">g1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c4">c3</font><br/>  for <font color="Olive" title="b3">g2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c6">c5</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c6">c5</font>) <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; <font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">g2</font> holds <br/>(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c3">c2</font>) <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;  ex <font color="Olive" title="b4">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c3">c2</font> st <br/>( <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g1</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font> &amp; (  for <font color="Olive" title="b5">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c3">c2</font>  st <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b5">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g1</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b5">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font> holds <br/><font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h1</font> ) ) ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</a></span>;<br/>
<div><span class="kw">hereby </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="c2">c1</font>,<font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c12">f5</font>,<font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font> implies <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c11">f4</font>,<font color="Maroon" title="c13">f6</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:75_1"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c11">f4</font>,<font color="Maroon" title="c13">f6</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="c2">c1</font>,<font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c12">f5</font>,<font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font> )</span><br/><a NAME="E2:75_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A5</font></span>: 
( <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font> &amp; (  for <font color="Olive" title="b1">d1</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b2">g1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c3">c2</font><br/>  for <font color="Olive" title="b3">g2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c5">c4</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c3">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c5">c4</font>) <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; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">g2</font> holds <br/>(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b4">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font> &amp; (  for <font color="Olive" title="b5">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c2">c1</font>  st <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b5">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b5">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font> holds <br/><font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h1</font> ) ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</a></span>;<br/><a NAME="E3:75_1"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c7">c6</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c5">c4</font>) <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="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#T22" target="_self" onmouseover="rs('cat_7/T22')" onmouseout="rh()">Th22</a></span>;<br/><span class="lab"><font color="Green" title="E47">A7</font></span>: <font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
<font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E2:75_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a>, <a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>
;<br/><a NAME="E5:75_1"/>
 for <font color="Olive" title="b1">d1</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b2">g1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c4">c3</font><br/>  for <font color="Olive" title="b3">g2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c5">c4</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c5">c4</font>) <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; <font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">g2</font> holds <br/>(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b4">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font> &amp; (  for <font color="Olive" title="b5">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c2">c1</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b5">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b5">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font> holds <br/><font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h1</font> ) ) )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="75_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">d1</font> be   <a href="cat_6.html#NM3" title="CAT_6:NM.3">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">  for <font color="Olive" title="b1">g1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c4">c3</font><br/>  for <font color="Olive" title="b2">g2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</font>) <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; <font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">g2</font> holds <br/>(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b3">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> &amp; (  for <font color="Olive" title="b4">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> holds <br/><font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">h1</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c16">g1</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c4">c3</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">  for <font color="Olive" title="b1">g2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</font>) <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; <font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">g2</font> holds <br/>(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b2">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g2</font> &amp; (  for <font color="Olive" title="b3">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g2</font> holds <br/><font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h1</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c17">g2</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</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"> (  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</font>) <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; <font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</font> implies (  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b2">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> ) ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:75_1_2"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c4">c3</font>) <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"> (  not  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</font> or (  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b2">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:75_1_2"/><span class="lab"><font color="Green" title="E49">A9</font></span>: 
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</font>) <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"> (  not <font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</font> or (  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b2">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:75_1_2"/><span class="lab"><font color="Green" title="E50">A10</font></span>: 
<font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</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"> (  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b2">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> ) ) )</span><br/>

<a NAME="E4:75_1_2"/><span class="lab"><font color="Green" title="E51">A11</font></span>: 
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c6">c5</font>) <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="#E2:75_1_2"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#T22" target="_self" onmouseover="rs('cat_7/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<a NAME="E5:75_1_2"/><span class="lab"><font color="Green" title="E52">A12</font></span>: 
<font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1_2"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="txt" href="#E2:75_1_2"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<a NAME="E6:75_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A13</font></span>: 
(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c3">c2</font>) <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;  ex <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c3">c2</font> st <br/>( <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b2">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c3">c2</font>  st <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E4:75_1_2"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E1:75_1_2"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">g3</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c3">c2</font><span class="kw"> such that </span><br/><a NAME="E8:75_1_2"/><span class="lab"><font color="Green" title="E54">A14</font></span>: 
( <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c18">g3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c18">g3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c3">c2</font>  st <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Maroon" title="c18">g3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E5:75_1_2"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E4:75_1_2"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E1:75_1_2"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:75_1_2"/><span class="lab"><font color="Green" title="E55">A15</font></span>: 
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E6:75_1_2"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="txt" href="#E2:75_1_2"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="txt" href="#E1:75_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</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">  ex <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b2">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c19">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font><span class="kw"> such that </span><br/><a NAME="E11:75_1_2"/><span class="lab"><font color="Green" title="E56">A16</font></span>: 
( <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g3</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g3</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E8:75_1_2"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="txt" href="#E6:75_1_2"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="txt" href="#E2:75_1_2"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="txt" href="#E1:75_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c19">h</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">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E12:75_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E8:75_1_2"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="txt" href="#E11:75_1_2"><span class="lab"><font color="Green" title="E56">A16</font></span></a>, <a class="txt" href="#E9:75_1_2"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</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="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E13:75_1_2"/>
<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:75_1_2"><span class="lab"><font color="Green" title="E56">A16</font></span></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">  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c20">h1</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</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">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c20">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c20">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> implies <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E14:75_1_2"/><span class="lab"><font color="Green" title="E57">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c20">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</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"> (  not <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c20">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> or <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E15:75_1_2"/><span class="lab"><font color="Green" title="E58">A18</font></span>: 
<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c20">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</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="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h1</font></span><br/>

<a NAME="E16:75_1_2"/><span class="lab"><font color="Green" title="E59">A19</font></span>: 
<font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E14:75_1_2"><span class="lab"><font color="Green" title="E57">A17</font></span></a>, <a class="txt" href="#E9:75_1_2"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c20">h1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c20">h1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E9:75_1_2"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
<font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:75_1_2"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="txt" href="#E2:75_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E9:75_1_2"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a>, <a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a></span>
;<br/>
<a NAME="E18:75_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">g3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c20">h1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:75_1_2"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="txt" href="#E8:75_1_2"><span class="lab"><font color="Green" title="E54">A14</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:75_1_2"/>
<font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:75_1_2"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="txt" href="#E11:75_1_2"><span class="lab"><font color="Green" title="E56">A16</font></span></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"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:75_1"/>
<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c12">f5</font>,<font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E3:75_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E4:75_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</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="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font></span><br/><span class="kw">thus </span><a NAME="E7:75_1"/>
<font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E1:75_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</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"> verum</span><br/></div>
<span class="kw">end;</span></div>

<a NAME="E5:75"/><span class="lab"><font color="Green" title="E44">A20</font></span>: 
(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c7">c6</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c7">c6</font>) <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="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#T22" target="_self" onmouseover="rs('cat_7/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:75"/><span class="lab"><font color="Green" title="E45">A21</font></span>: 
<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c12">f5</font>,<font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</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"> (  not <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</font> or <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c11">f4</font>,<font color="Maroon" title="c13">f6</font> )</span><br/>

<span class="kw">assume </span><a NAME="E7:75"/><span class="lab"><font color="Green" title="E46">A22</font></span>: 
<font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f3</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="c2">c1</font>,<font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c11">f4</font>,<font color="Maroon" title="c13">f6</font></span><br/>

<a NAME="E8:75"/>
 for <font color="Olive" title="b1">d1</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b2">g1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c3">c2</font><br/>  for <font color="Olive" title="b3">g2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c5">c4</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c3">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c5">c4</font>) <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; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">g2</font> holds <br/>(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b4">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font> &amp; (  for <font color="Olive" title="b5">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c2">c1</font>  st <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b5">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b5">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font> holds <br/><font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h1</font> ) ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="75_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">d1</font> be   <a href="cat_6.html#NM3" title="CAT_6:NM.3">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">  for <font color="Olive" title="b1">g1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c3">c2</font><br/>  for <font color="Olive" title="b2">g2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c3">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</font>) <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; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">g2</font> holds <br/>(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b3">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> &amp; (  for <font color="Olive" title="b4">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> holds <br/><font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">h1</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c16">g1</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c3">c2</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">  for <font color="Olive" title="b1">g2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c3">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</font>) <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; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">g2</font> holds <br/>(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b2">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g2</font> &amp; (  for <font color="Olive" title="b3">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g2</font> holds <br/><font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h1</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c17">g2</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</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"> (  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c3">c2</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</font>) <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; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</font> implies (  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b2">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> ) ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:75_2"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c3">c2</font>) <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"> (  not  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</font> or (  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b2">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:75_2"/><span class="lab"><font color="Green" title="E48">A24</font></span>: 
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c5">c4</font>) <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"> (  not <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</font> or (  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b2">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:75_2"/><span class="lab"><font color="Green" title="E49">A25</font></span>: 
<font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</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"> (  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b2">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> ) ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c18">g11</font> = <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font>;<br/>
<a NAME="E4:75_2"/><span class="lab"><font color="Green" title="E50">A26</font></span>: 
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c4">c3</font>) <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="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E1:75_2"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="ref" href="cat_7.html#T22" target="_self" onmouseover="rs('cat_7/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<span class="lab"><font color="Green" title="E51">A27</font></span>: <font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_2"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
<font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f6</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_2"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="txt" href="#E1:75_2"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a>, <a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c13">f6</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">g2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_2"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>
;<br/>
<a NAME="E6:75_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A28</font></span>: 
(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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;  ex <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b2">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:75_2"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E4:75_2"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E6:75"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</a></span>;<br/>
<span class="kw">thus </span><a NAME="E7:75_2"/><span class="lab"><font color="Green" title="E53">A29</font></span>: 
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>) <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="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E5:75_2"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E2:75_2"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E4:75_2"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E6:75"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</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">  ex <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font> st <br/>( <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b2">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c19">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font><span class="kw"> such that </span><br/><a NAME="E9:75_2"/><span class="lab"><font color="Green" title="E54">A30</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E5:75_2"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E2:75_2"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E4:75_2"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E6:75"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c19">h</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="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c20">g22</font> = <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font>;<br/>
<a NAME="E10:75_2"/><span class="lab"><font color="Green" title="E55">A31</font></span>: 
(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c4">c3</font>) <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;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c6">c5</font>) <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="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E1:75_2"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="ref" href="cat_7.html#T22" target="_self" onmouseover="rs('cat_7/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<span class="lab"><font color="Green" title="E56">A32</font></span>: <font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f5</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_2"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
<font color="Maroon" title="c14">f7</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_2"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a>, <a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">h2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c3">c2</font><span class="kw"> such that </span><br/><a NAME="E13:75_2"/><span class="lab"><font color="Green" title="E57">A33</font></span>: 
( <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c21">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c21">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> &amp; (  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c3">c2</font>  st <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font> holds <br/><font color="Maroon" title="c21">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E11:75_2"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="txt" href="#E10:75_2"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</a></span>;<br/>
<a NAME="E14:75_2"/><span class="lab"><font color="Green" title="E58">A34</font></span>: 
<font color="Maroon" title="c21">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:75_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a></span>;<br/>
<a NAME="E15:75_2"/><span class="lab"><font color="Green" title="E59">A35</font></span>: 
<font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E9:75_2"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E6:75_2"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E6:75_2"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
<font color="Maroon" title="c11">f4</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:75_2"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E3:75_2"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="txt" href="#E7:75"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E6:75_2"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E17:75_2"/>
<font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:75_2"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="txt" href="#E15:75_2"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E14:75_2"><span class="lab"><font color="Green" title="E58">A34</font></span></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="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> &amp; (  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E18:75_2"/>
<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:75_2"><span class="lab"><font color="Green" title="E54">A30</font></span></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">  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</font>  st <font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> holds <br/><font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c22">h1</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c15">d1</font>,<font color="Maroon" title="c2">c1</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="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c22">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c22">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> implies <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E19:75_2"/><span class="lab"><font color="Green" title="E60">A36</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c22">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</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"> (  not <font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c22">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> or <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h1</font> )</span><br/>

<a NAME="E20:75_2"/><span class="lab"><font color="Green" title="E61">A37</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c22">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E19:75_2"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E7:75_2"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<span class="kw">assume </span><a NAME="E21:75_2"/>
<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c22">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</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="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h1</font></span><br/>

<span class="kw">hence </span><a NAME="E22:75_2"/>
<font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:75_2"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E20:75_2"><span class="lab"><font color="Green" title="E61">A37</font></span></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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:75"/>
<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c10">f3</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c11">f4</font>,<font color="Maroon" title="c13">f6</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</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"> verum</span><br/>


</div>
