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<b>let </b><font color="Maroon">F</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>





<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">F</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E2:30_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">F</font> <a href="zf_lang.html#R2">is_subformula_of</a> <font color="Maroon">G</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T83">ZF_LANG:83</a>, <a class="ref" href="zf_lang.html#D41">ZF_LANG:def 41</a></i>;<br/><b>assume </b><a NAME="E3:30_1"/>
<font color="Maroon">G</font> <a href="zf_lang.html#R2">is_subformula_of</a> <font color="Maroon">H</font>
 ;<br/><a NAME="E4:30_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="zf_lang.html#R2">is_subformula_of</a> <font color="Maroon">H</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> )
 <b>by </b><i>, , <a class="ref" href="zf_lang.html#T86">ZF_LANG:86</a></i>;<br/><b>hence </b><a NAME="E5:30_1"/>
<font color="Maroon">F</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="zf_lang.html#D41">ZF_LANG:def 41</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">F</font> <a href="zf_lang.html#R2">is_subformula_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">H</font> )
 ;<br/><a NAME="E2:30_2"/><b>then </b>
( <font color="Maroon">G</font> <a href="zf_lang.html#R2">is_subformula_of</a> <font color="Maroon">H</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="zf_lang.html#T83">ZF_LANG:83</a>, <a class="ref" href="zf_lang.html#D41">ZF_LANG:def 41</a></i>;<br/><a NAME="E3:30_2"/><b>then </b>
( <font color="Maroon">F</font> <a href="zf_lang.html#R2">is_subformula_of</a> <font color="Maroon">H</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="zf_lang.html#T86">ZF_LANG:86</a></i>;<br/><b>hence </b><a NAME="E4:30_2"/>
( ( ( <font color="Maroon">F</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R2">is_subformula_of</a> <font color="Maroon">H</font> ) or ( <font color="Maroon">F</font> <a href="zf_lang.html#R2">is_subformula_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">H</font> ) or ( <font color="Maroon">F</font> <a href="zf_lang.html#R2">is_subformula_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R1">is_immediate_constituent_of</a> <font color="Maroon">H</font> ) or ( <font color="Maroon">F</font> <a href="zf_lang.html#R1">is_immediate_constituent_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R2">is_subformula_of</a> <font color="Maroon">H</font> ) or ( <font color="Maroon">F</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R1">is_immediate_constituent_of</a> <font color="Maroon">H</font> ) or ( <font color="Maroon">F</font> <a href="zf_lang.html#R1">is_immediate_constituent_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">H</font> ) ) implies <font color="Maroon">F</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#D41">ZF_LANG:def 41</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:30"/>
( ( <font color="Maroon">G</font> <a href="zf_lang.html#R1">is_immediate_constituent_of</a> <font color="Maroon">H</font> implies <font color="Maroon">G</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">H</font> ) &amp; ( <font color="Maroon">F</font> <a href="zf_lang.html#R1">is_immediate_constituent_of</a> <font color="Maroon">G</font> implies <font color="Maroon">F</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">G</font> ) )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T82">ZF_LANG:82</a></i>;<br/>
<b>hence </b><a NAME="E4:30"/>
( ( ( <font color="Maroon">F</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R2">is_subformula_of</a> <font color="Maroon">H</font> ) or ( <font color="Maroon">F</font> <a href="zf_lang.html#R2">is_subformula_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">H</font> ) or ( <font color="Maroon">F</font> <a href="zf_lang.html#R2">is_subformula_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R1">is_immediate_constituent_of</a> <font color="Maroon">H</font> ) or ( <font color="Maroon">F</font> <a href="zf_lang.html#R1">is_immediate_constituent_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R2">is_subformula_of</a> <font color="Maroon">H</font> ) or ( <font color="Maroon">F</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R1">is_immediate_constituent_of</a> <font color="Maroon">H</font> ) or ( <font color="Maroon">F</font> <a href="zf_lang.html#R1">is_immediate_constituent_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">H</font> ) ) implies <font color="Maroon">F</font> <a href="zf_lang.html#R3">is_proper_subformula_of</a> <font color="Maroon">H</font> )
 <b>by </b><i>, , <a class="ref" href="zf_lang.html#T85">ZF_LANG:85</a></i>;<br/>


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