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 $$ (Path_matrix (p,M)) ) & ( for p being Element of Permutations n holds f2 . p = the multF of K $$ (Path_matrix (p,M)) ) implies f1 = f2 )

assume that

A2: for p being Element of Permutations n holds f1 . p = the multF of K $$ (Path_matrix (p,M)) and

A3: for p being Element of Permutations n holds f2 . p = the multF of K $$ (Path_matrix (p,M)) ; :: thesis: f1 = f2

assume that

A2: for p being Element of Permutations n holds f1 . p = the multF of K $$ (Path_matrix (p,M)) and

A3: for p being Element of Permutations n holds f2 . p = the multF of K $$ (Path_matrix (p,M)) ; :: thesis: f1 = f2

now :: thesis: for p being Element of Permutations n holds f1 . p = f2 . p

hence
f1 = f2
by FUNCT_2:63; :: thesis: verumlet p be Element of Permutations n; :: thesis: f1 . p = f2 . p

f1 . p = the multF of K $$ (Path_matrix (p,M)) by A2;

hence f1 . p = f2 . p by A3; :: thesis: verum

end;f1 . p = the multF of K $$ (Path_matrix (p,M)) by A2;

hence f1 . p = f2 . p by A3; :: thesis: verum