deffunc H₁( Element of X, Element of X, Element of Y, Element of Y) -> Element of REAL = sqrt (((dist $1,$2) ^2 ) + ((dist $3,$4) ^2 ));
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 = sqrt (((dist x1,y1) ^2 ) + ((dist x2,y2) ^2 ))
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 = sqrt (((dist x1,y1) ^2 ) + ((dist x2,y2) ^2 ))
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 = sqrt (((dist x1,y1) ^2 ) + ((dist x2,y2) ^2 ))
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 = sqrt (((dist x1,y1) ^2 ) + ((dist x2,y2) ^2 )) )
assume ( x = [x1,x2] & y = [y1,y2] ) ; :: thesis: F . x,y = sqrt (((dist x1,y1) ^2 ) + ((dist x2,y2) ^2 ))
hence F . x,y = sqrt (((dist x1,y1) ^2 ) + ((dist x2,y2) ^2 )) by A1; :: thesis: verum