let f1, f2 be Function of (G1 .allSG()),(G2 .allSG()); :: thesis: ( ( for H being plain Subgraph of G1 holds f1 . H = rng (F | H) ) & ( for H being plain Subgraph of G1 holds f2 . H = rng (F | H) ) implies f1 = f2 )
assume that
A3: for H being plain Subgraph of G1 holds f1 . H = rng (F | H) and
A4: for H being plain Subgraph of G1 holds f2 . H = rng (F | H) ; :: thesis: f1 = f2

now :: thesis: for x being object st x in G1 .allSG() holds
f1 . x = f2 . x

let x be object ; :: thesis: ( x in G1 .allSG() implies f1 . x = f2 . x )
assume x in G1 .allSG() ; :: thesis: f1 . x = f2 . x
then reconsider H = x as plain Subgraph of G1 by Th1;
thus f1 . x = rng (F | H) by A3
.= f2 . x by A4 ; :: thesis: verum

end;

hence f1 = f2 by FUNCT_2:12; :: thesis: verum