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<span class="kw">set </span><font color="Maroon" title="c3">gf</font> = <font color="Maroon" title="c2">g</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c1">f</font>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_1"><span class="kw">now </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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c1">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> is   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c1">f</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> is   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )</span><br/><a NAME="E1:71_1_1_1"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#D6" onmouseover="rs('fomodel2/D6')" onmouseout="rh()">FOMODEL2:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">X</font> = <font color="Maroon" title="c4">x</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/><span class="kw">assume </span><a NAME="E3:71_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c1">f</font></span>)</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> is   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/><a NAME="E4:71_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="fomodel2.html#D6" onmouseover="rs('fomodel2/D6')" onmouseout="rh()">FOMODEL2:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E5:71_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> is   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:71_1_1"/>
<font color="Maroon" title="c2">g</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c1">f</font> is  <a href="valued_2.html#V12" title="VALUED_2:attr.12">natural-functions-valued</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_2.html#D31" onmouseover="rs('valued_2/D31')" onmouseout="rh()">VALUED_2:def 31</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/>


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