deffunc H₁( Element of REAL , Element of REAL , Element of REAL , Element of REAL ) -> Element of REAL = (real_dist . $1,$2) + (real_dist . $3,$4);
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 = (real_dist . x1,y1) + (real_dist . x2,y2)
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 = (real_dist . x1,y1) + (real_dist . x2,y2)
let x, y be Element of [:REAL ,REAL :]; :: thesis: ( x = [x1,x2] & y = [y1,y2] implies F . x,y = (real_dist . x1,y1) + (real_dist . x2,y2) )
assume A2: ( x = [x1,x2] & y = [y1,y2] ) ; :: thesis: F . x,y = (real_dist . x1,y1) + (real_dist . x2,y2)
thus F . x,y = (real_dist . x1,y1) + (real_dist . x2,y2) by A1, A2; :: thesis: verum