let G be _Graph; :: thesis: for W1, W2, W3 being Walk of G holds
( W1 .first() = (W1 .replaceWith (W2,W3)) .first() & W1 .last() = (W1 .replaceWith (W2,W3)) .last() )
let W1, W2, W3 be Walk of G; :: thesis: ( W1 .first() = (W1 .replaceWith (W2,W3)) .first() & W1 .last() = (W1 .replaceWith (W2,W3)) .last() )

per cases ( ( W2 is_odd_substring_of W1, 0 & W2 .first() = W3 .first() & W2 .last() = W3 .last() ) or not W2 is_odd_substring_of W1, 0 or not W2 .first() = W3 .first() or not W2 .last() = W3 .last() ) ;

suppose A1: ( W2 is_odd_substring_of W1, 0 & W2 .first() = W3 .first() & W2 .last() = W3 .last() ) ; :: thesis: ( W1 .first() = (W1 .replaceWith (W2,W3)) .first() & W1 .last() = (W1 .replaceWith (W2,W3)) .last() )

then A2: W1 .replaceWith (W2,W3) = ((W1 .cut (1,(W1 .findFirstVertex W2))) .append W3) .append (W1 .cut ((W1 .findLastVertex W2),(len W1))) by Def5;
reconsider n1 = 1 as odd Element of NAT by POLYFORM:4;
set n2 = W1 .findFirstVertex W2;
set n3 = W1 .findLastVertex W2;
set n4 = len W1;
set W1c = W1 .cut (n1,(W1 .findFirstVertex W2));
set W1ca = (W1 .cut (n1,(W1 .findFirstVertex W2))) .append W3;
set W1c2 = W1 .cut ((W1 .findLastVertex W2),(len W1));
set W1caa = ((W1 .cut (n1,(W1 .findFirstVertex W2))) .append W3) .append (W1 .cut ((W1 .findLastVertex W2),(len W1)));
( ((W1 .cut (n1,(W1 .findFirstVertex W2))) .append W3) .last() = W3 .last() & (W1 .cut ((W1 .findLastVertex W2),(len W1))) .first() = W3 .last() ) by A1, Th38;
then ( (((W1 .cut (n1,(W1 .findFirstVertex W2))) .append W3) .append (W1 .cut ((W1 .findLastVertex W2),(len W1)))) .first() = ((W1 .cut (n1,(W1 .findFirstVertex W2))) .append W3) .first() & (((W1 .cut (n1,(W1 .findFirstVertex W2))) .append W3) .append (W1 .cut ((W1 .findLastVertex W2),(len W1)))) .last() = (W1 .cut ((W1 .findLastVertex W2),(len W1))) .last() ) by GLIB_001:30;
hence ( W1 .first() = (W1 .replaceWith (W2,W3)) .first() & W1 .last() = (W1 .replaceWith (W2,W3)) .last() ) by A1, A2, Th38; :: thesis: verum

end;

suppose ( not W2 is_odd_substring_of W1, 0 or not W2 .first() = W3 .first() or not W2 .last() = W3 .last() ) ; :: thesis: ( W1 .first() = (W1 .replaceWith (W2,W3)) .first() & W1 .last() = (W1 .replaceWith (W2,W3)) .last() )

hence ( W1 .first() = (W1 .replaceWith (W2,W3)) .first() & W1 .last() = (W1 .replaceWith (W2,W3)) .last() ) by Def5; :: thesis: verum

end;

end;