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<span class="kw">let </span><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font> ) ) implies <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font> )</span><br/>





<a NAME="E1:34"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">G</font> implies <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T82" title="ZF_LANG:th.82">ZF_LANG:82</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E13">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:34_1"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> implies <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font> )</span><br/><a NAME="E2:34_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T83" title="ZF_LANG:th.83">ZF_LANG:83</a></span>;<br/><span class="kw">assume </span><a NAME="E3:34_1"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font></span><br/><a NAME="E4:34_1"/>
<font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="zf_lang.html#D41" title="ZF_LANG:def.41">ZF_LANG:def 41</a></span>;<br/><a NAME="E5:34_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="zf_lang.html#T86" title="ZF_LANG:th.86">ZF_LANG:86</a></span>;<br/><a NAME="E6:34_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="zf_lang1.html#T43" target="_self" title="ZF_LANG1:th.43">Th43</a></span>;<br/><span class="kw">hence </span><a NAME="E7:34_1"/>
<font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E5:34_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="zf_lang.html#D41" title="ZF_LANG:def.41">ZF_LANG:def 41</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E14">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34_2"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E2:34_2"/><span class="lab"><font color="Green" title="E16">A9</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font> ) ) implies <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font> )</span><br/><a NAME="E3:34_2"/>
<font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="zf_lang.html#D41" title="ZF_LANG:def.41">ZF_LANG:def 41</a></span>;<br/><a NAME="E4:34_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A10</font></span>: 
<font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="zf_lang.html#T86" title="ZF_LANG:th.86">ZF_LANG:86</a></span>;<br/><a NAME="E5:34_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="zf_lang1.html#T43" target="_self" title="ZF_LANG1:th.43">Th43</a></span>;<br/><a NAME="E6:34_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_2"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="zf_lang.html#T83" title="ZF_LANG:th.83">ZF_LANG:83</a></span>;<br/><span class="kw">hence </span><a NAME="E7:34_2"/>
( ( ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font> ) ) implies <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="ref" href="zf_lang.html#D41" title="ZF_LANG:def.41">ZF_LANG:def 41</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:34"/>
( <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c3">H</font> implies <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T82" title="ZF_LANG:th.82">ZF_LANG:82</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:34"/>
( ( ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R2" title="ZF_LANG:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c3">H</font> ) or ( <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c2">G</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font> ) ) implies <font color="Maroon" title="c1">F</font> <a href="zf_lang.html#R3" title="ZF_LANG:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c3">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="zf_lang.html#T85" title="ZF_LANG:th.85">ZF_LANG:85</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
