let IT1, IT2 be FF:ELabelingSeq of G; :: thesis: ( IT1 . 0 = (the_Edges_of G) --> 0 & ( for n being Nat holds IT1 . (n + 1) = FF:Step (IT1 . n),source,sink ) & IT2 . 0 = (the_Edges_of G) --> 0 & ( for n being Nat holds IT2 . (n + 1) = FF:Step (IT2 . n),source,sink ) implies IT1 = IT2 )
assume that
A15: ( IT1 . 0 = (the_Edges_of G) --> 0 & ( for n being Nat holds IT1 . (n + 1) = FF:Step (IT1 . n),source,sink ) ) and
A16: ( IT2 . 0 = (the_Edges_of G) --> 0 & ( for n being Nat holds IT2 . (n + 1) = FF:Step (IT2 . n),source,sink ) ) ; :: thesis: IT1 = IT2
defpred S₁[ Nat] means IT1 . $1 = IT2 . $1;
A17: S₁[ 0 ] by A15, A16;

A18: now

let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A19: S₁[n] ; :: thesis: S₁[n + 1]
A21: IT2 . (n + 1) = FF:Step (IT2 . n),source,sink by A16;
thus S₁[n + 1] by A19, A15, A21; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A17, A18);
then for n being set st n in NAT holds
IT1 . n = IT2 . n ;
hence IT1 = IT2 by PBOOLE:3; :: thesis: verum