let X be set ; :: thesis: for f being Function of [:X,X:],REAL holds
( f is_a_pseudometric_of X iff for a, b, c being Element of X holds
( f . (a,a) = 0 & f . (a,c) <= (f . (a,b)) + (f . (c,b)) ) )
let f be Function of [:X,X:],REAL; :: thesis: ( f is_a_pseudometric_of X iff for a, b, c being Element of X holds
( f . (a,a) = 0 & f . (a,c) <= (f . (a,b)) + (f . (c,b)) ) )
thus ( f is_a_pseudometric_of X implies for a, b, c being Element of X holds
( f . (a,a) = 0 & f . (a,c) <= (f . (a,b)) + (f . (c,b)) ) ) :: thesis: ( ( for a, b, c being Element of X holds
( f . (a,a) = 0 & f . (a,c) <= (f . (a,b)) + (f . (c,b)) ) ) implies f is_a_pseudometric_of X )

proof

assume A1: f is_a_pseudometric_of X ; :: thesis: for a, b, c being Element of X holds
( f . (a,a) = 0 & f . (a,c) <= (f . (a,b)) + (f . (c,b)) )
let a, b, c be Element of X; :: thesis: ( f . (a,a) = 0 & f . (a,c) <= (f . (a,b)) + (f . (c,b)) )
f . (a,c) <= (f . (a,b)) + (f . (b,c)) by A1, Lm8;
hence ( f . (a,a) = 0 & f . (a,c) <= (f . (a,b)) + (f . (c,b)) ) by A1, Lm8; :: thesis: verum

end;

thus ( ( for a, b, c being Element of X holds
( f . (a,a) = 0 & f . (a,c) <= (f . (a,b)) + (f . (c,b)) ) ) implies f is_a_pseudometric_of X ) :: thesis: verum

proof

assume A2: for a, b, c being Element of X holds
( f . (a,a) = 0 & f . (a,c) <= (f . (a,b)) + (f . (c,b)) ) ; :: thesis: f is_a_pseudometric_of X
A3: for a, b being Element of X holds f . (a,b) = f . (b,a)

proof

let a, b be Element of X; :: thesis: f . (a,b) = f . (b,a)
A4: ( f . (b,a) <= (f . (b,b)) + (f . (a,b)) & f . (b,b) = 0 ) by A2;
( f . (a,b) <= (f . (a,a)) + (f . (b,a)) & f . (a,a) = 0 ) by A2;
hence f . (a,b) = f . (b,a) by A4, XXREAL_0:1; :: thesis: verum

end;

for a, b, c being Element of X holds f . (a,c) <= (f . (a,b)) + (f . (b,c))

proof

let x, y, z be Element of X; :: thesis: f . (x,z) <= (f . (x,y)) + (f . (y,z))
f . (x,z) <= (f . (x,y)) + (f . (z,y)) by A2;
hence f . (x,z) <= (f . (x,y)) + (f . (y,z)) by A3; :: thesis: verum

end;

then ( f is Reflexive & f is symmetric & f is triangle ) by A2, A3, METRIC_1:def 2, METRIC_1:def 4, METRIC_1:def 5;
hence f is_a_pseudometric_of X ; :: thesis: verum

end;