deffunc H₁( Element of REAL , Element of REAL , Element of REAL , Element of REAL ) -> Element of REAL = sqrt (((real_dist . $1,$2) ^2 ) + ((real_dist . $3,$4) ^2 ));
consider F being Function of [:[:REAL ,REAL :],[:REAL ,REAL :]:],REAL such that
A1: for x1, y1, x2, y2 being Element of REAL
for x, y being Element of [:REAL ,REAL :] 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, x2, y2 being Element of REAL
for x, y being Element of [:REAL ,REAL :] st x = [x1,x2] & y = [y1,y2] holds
F . x,y = sqrt (((real_dist . x1,y1) ^2 ) + ((real_dist . x2,y2) ^2 ))
let x1, y1, x2, y2 be Element of REAL ; :: thesis: for x, y being Element of [:REAL ,REAL :] st x = [x1,x2] & y = [y1,y2] holds
F . x,y = sqrt (((real_dist . x1,y1) ^2 ) + ((real_dist . x2,y2) ^2 ))
let x, y be Element of [:REAL ,REAL :]; :: thesis: ( x = [x1,x2] & y = [y1,y2] implies F . x,y = sqrt (((real_dist . x1,y1) ^2 ) + ((real_dist . x2,y2) ^2 )) )
assume ( x = [x1,x2] & y = [y1,y2] ) ; :: thesis: F . x,y = sqrt (((real_dist . x1,y1) ^2 ) + ((real_dist . x2,y2) ^2 ))
hence F . x,y = sqrt (((real_dist . x1,y1) ^2 ) + ((real_dist . x2,y2) ^2 )) by A1; :: thesis: verum