let f1, f2 be Function of [:[:REAL,REAL:],[:REAL,REAL:]:],REAL; :: 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
f1 . (x,y) = sqrt (((real_dist . (x1,y1)) ^2) + ((real_dist . (x2,y2)) ^2)) ) & ( 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
f2 . (x,y) = sqrt (((real_dist . (x1,y1)) ^2) + ((real_dist . (x2,y2)) ^2)) ) implies f1 = f2 )
assume that
A2: 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
f1 . (x,y) = sqrt (((real_dist . (x1,y1)) ^2) + ((real_dist . (x2,y2)) ^2)) and
A3: 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
f2 . (x,y) = sqrt (((real_dist . (x1,y1)) ^2) + ((real_dist . (x2,y2)) ^2)) ; :: thesis: f1 = f2
for x, y being Element of [:REAL,REAL:] holds f1 . (x,y) = f2 . (x,y)

proof

let x, y be Element of [:REAL,REAL:]; :: thesis: f1 . (x,y) = f2 . (x,y)
consider x1, x2 being Element of REAL such that
A4: x = [x1,x2] by DOMAIN_1:1;
consider y1, y2 being Element of REAL such that
A5: y = [y1,y2] by DOMAIN_1:1;
thus f1 . (x,y) = sqrt (((real_dist . (x1,y1)) ^2) + ((real_dist . (x2,y2)) ^2)) by A2, A4, A5
.= f2 . (x,y) by A3, A4, A5 ; :: thesis: verum

end;

hence f1 = f2 by BINOP_1:2; :: thesis: verum