let f1, f2 be Function of (Permutations n),the carrier of K; :: thesis: ( ( for p being Element of Permutations n holds f1 . p = - (the multF of K $$ (Path_matrix p,M)),p ) & ( for p being Element of Permutations n holds f2 . p = - (the multF of K $$ (Path_matrix p,M)),p ) implies f1 = f2 )
assume that
A2: for p being Element of Permutations n holds f1 . p = - (the multF of K $$ (Path_matrix p,M)),p and
A3: for p being Element of Permutations n holds f2 . p = - (the multF of K $$ (Path_matrix p,M)),p ; :: thesis: f1 = f2

now

let p be Element of Permutations n; :: thesis: f1 . p = f2 . p
f1 . p = - (the multF of K $$ (Path_matrix p,M)),p by A2;
hence f1 . p = f2 . p by A3; :: thesis: verum

end;

hence f1 = f2 by FUNCT_2:113; :: thesis: verum