deffunc H₁( Element of X, Element of X, Element of Y, Element of Y, Element of Z, Element of Z) -> Element of REAL = ((dist $1,$2) + (dist $3,$4)) + (dist $5,$6);
consider F being Function of [:[:the carrier of X,the carrier of Y,the carrier of Z:],[:the carrier of X,the carrier of Y,the carrier of Z:]:],REAL such that
A1: for x1, y1 being Element of X
for x2, y2 being Element of Y
for x3, y3 being Element of Z
for x, y being Element of [:the carrier of X,the carrier of Y,the carrier of Z:] st x = [x1,x2,x3] & y = [y1,y2,y3] holds
F . x,y = H₁(x1,y1,x2,y2,x3,y3) from METRIC_3:sch 2();
take F ; :: thesis: for x1, y1 being Element of X
for x2, y2 being Element of Y
for x3, y3 being Element of Z
for x, y being Element of [:the carrier of X,the carrier of Y,the carrier of Z:] st x = [x1,x2,x3] & y = [y1,y2,y3] holds
F . x,y = ((dist x1,y1) + (dist x2,y2)) + (dist x3,y3)
let x1, y1 be Element of X; :: thesis: for x2, y2 being Element of Y
for x3, y3 being Element of Z
for x, y being Element of [:the carrier of X,the carrier of Y,the carrier of Z:] st x = [x1,x2,x3] & y = [y1,y2,y3] holds
F . x,y = ((dist x1,y1) + (dist x2,y2)) + (dist x3,y3)
let x2, y2 be Element of Y; :: thesis: for x3, y3 being Element of Z
for x, y being Element of [:the carrier of X,the carrier of Y,the carrier of Z:] st x = [x1,x2,x3] & y = [y1,y2,y3] holds
F . x,y = ((dist x1,y1) + (dist x2,y2)) + (dist x3,y3)
let x3, y3 be Element of Z; :: thesis: for x, y being Element of [:the carrier of X,the carrier of Y,the carrier of Z:] st x = [x1,x2,x3] & y = [y1,y2,y3] holds
F . x,y = ((dist x1,y1) + (dist x2,y2)) + (dist x3,y3)
let x, y be Element of [:the carrier of X,the carrier of Y,the carrier of Z:]; :: thesis: ( x = [x1,x2,x3] & y = [y1,y2,y3] implies F . x,y = ((dist x1,y1) + (dist x2,y2)) + (dist x3,y3) )
assume A2: ( x = [x1,x2,x3] & y = [y1,y2,y3] ) ; :: thesis: F . x,y = ((dist x1,y1) + (dist x2,y2)) + (dist x3,y3)
thus F . x,y = ((dist x1,y1) + (dist x2,y2)) + (dist x3,y3) by A1, A2; :: thesis: verum