deffunc H₁( Element of X, Element of X, Element of Y, Element of Y) -> Element of REAL = In (((dist ($1,$2)) + (dist ($3,$4))),REAL);
consider F being Function of [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],REAL such that
A1: for x1, y1 being Element of X
for x2, y2 being Element of Y
for x, y being Element of [: the carrier of X, the carrier of Y:] st x = [x1,x2] & y = [y1,y2] holds
F . (x,y) = H₁(x1,y1,x2,y2) from METRIC_3:sch 1();
take F ; :: thesis: for x1, y1 being Element of X
for x2, y2 being Element of Y
for x, y being Element of [: the carrier of X, the carrier of Y:] st x = [x1,x2] & y = [y1,y2] holds
F . (x,y) = (dist (x1,y1)) + (dist (x2,y2))
let x1, y1 be Element of X; :: thesis: for x2, y2 being Element of Y
for x, y being Element of [: the carrier of X, the carrier of Y:] st x = [x1,x2] & y = [y1,y2] holds
F . (x,y) = (dist (x1,y1)) + (dist (x2,y2))
let x2, y2 be Element of Y; :: thesis: for x, y being Element of [: the carrier of X, the carrier of Y:] st x = [x1,x2] & y = [y1,y2] holds
F . (x,y) = (dist (x1,y1)) + (dist (x2,y2))
let x, y be Element of [: the carrier of X, the carrier of Y:]; :: thesis: ( x = [x1,x2] & y = [y1,y2] implies F . (x,y) = (dist (x1,y1)) + (dist (x2,y2)) )
assume ( x = [x1,x2] & y = [y1,y2] ) ; :: thesis: F . (x,y) = (dist (x1,y1)) + (dist (x2,y2))
then F . (x,y) = H₁(x1,y1,x2,y2) by A1;
hence F . (x,y) = (dist (x1,y1)) + (dist (x2,y2)) ; :: thesis: verum