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
A8: IT1 . 0 = (the_Edges_of G) --> 0 and
A9: for n being Nat holds IT1 . (n + 1) = FF:Step ((IT1 . n),source,sink) and
A10: IT2 . 0 = (the_Edges_of G) --> 0 and
A11: 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;

A12: now :: thesis: for n being Nat st S₁[n] holds
S₁[n + 1]

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A13: S₁[n] ; :: thesis: S₁[n + 1]
IT2 . (n + 1) = FF:Step ((IT2 . n),source,sink) by A11;
hence S₁[n + 1] by A9, A13; :: thesis: verum

end;

A14: S₁[ 0 ] by A8, A10;
for n being Nat holds S₁[n] from NAT_1:sch 2(A14, A12);
then for n being object st n in NAT holds
IT1 . n = IT2 . n ;
hence IT1 = IT2 by PBOOLE:3; :: thesis: verum