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<span class="kw">let </span><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">p</font> is  <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a>  &amp; <font color="Maroon" title="c2">q</font> is  <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a>  ) iff <font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">q</font> is  <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a>  )</span><br/>



<a NAME="E1:55"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T31" title="FINSEQ_1:th.31">FINSEQ_1:31</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:55"/>
( <font color="Maroon" title="c1">p</font> is  <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a>  &amp; <font color="Maroon" title="c2">q</font> is  <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a>  implies <font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">q</font> is  <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">q</font> is  <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a>  implies ( <font color="Maroon" title="c1">p</font> is  <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a>  &amp; <font color="Maroon" title="c2">q</font> is  <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:55_1"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font> is  <a href="trees_3.html#V1" title="TREES_3:attr.1">constituted-Trees</a> 
 <span class="kw">and </span><br/><a NAME="E2:55_1"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">q</font> is  <a href="trees_3.html#V1" title="TREES_3:attr.1">constituted-Trees</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="trees_3.html#D9" target="_self" title="TREES_3:def.9">TREES_3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">q</font> is  <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a> </span><br/>

<span class="kw">thus </span><a NAME="E3:55_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">q</font></span>)</span> is  <a href="trees_3.html#V1" title="TREES_3:attr.1">constituted-Trees</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E1:55_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="trees_3.html#T3" target="_self" title="TREES_3:th.3">Th3</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="trees_3.html#D9" target="_self" title="TREES_3:def.9">TREES_3:def 9</a> <a class="txt" 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">assume </span><a NAME="E3:55"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c2">q</font></span>)</span> is  <a href="trees_3.html#V1" title="TREES_3:attr.1">constituted-Trees</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="trees_3.html#D9" target="_self" title="TREES_3:def.9">TREES_3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> is  <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a>  &amp; <font color="Maroon" title="c2">q</font> is  <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a>  )</span><br/>

<span class="kw">hence </span><a NAME="E4:55"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font> is  <a href="trees_3.html#V1" title="TREES_3:attr.1">constituted-Trees</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="trees_3.html#T3" target="_self" title="TREES_3:th.3">Th3</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="trees_3.html#D9" target="_self" title="TREES_3:def.9">TREES_3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">q</font> is  <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a> </span><br/>

<span class="kw">thus </span><a NAME="E5:55"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">q</font> is  <a href="trees_3.html#V1" title="TREES_3:attr.1">constituted-Trees</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="trees_3.html#T3" target="_self" title="TREES_3:th.3">Th3</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="trees_3.html#D9" target="_self" title="TREES_3:def.9">TREES_3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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