let X, Y be non empty MetrSpace; for x, y being Element of [: the carrier of X, the carrier of Y:] holds
( (dist_cart2S (X,Y)) . (x,y) = 0 iff x = y )
let x, y be Element of [: the carrier of X, the carrier of Y:]; ( (dist_cart2S (X,Y)) . (x,y) = 0 iff x = y )
reconsider x1 = x `1 , y1 = y `1 as Element of X ;
reconsider x2 = x `2 , y2 = y `2 as Element of Y ;
A1:
( x = [x1,x2] & y = [y1,y2] )
;
thus
( (dist_cart2S (X,Y)) . (x,y) = 0 implies x = y )
( x = y implies (dist_cart2S (X,Y)) . (x,y) = 0 )proof
set d2 =
dist (
x2,
y2);
set d1 =
dist (
x1,
y1);
assume
(dist_cart2S (X,Y)) . (
x,
y)
= 0
;
x = y
then A2:
sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2)) = 0
by A1, Def10;
A3:
(
0 <= (dist (x1,y1)) ^2 &
0 <= (dist (x2,y2)) ^2 )
by XREAL_1:63;
then
dist (
x1,
y1)
= 0
by A2, Lm1;
then A4:
x1 = y1
by METRIC_1:2;
dist (
x2,
y2)
= 0
by A2, A3, Lm1;
hence
x = y
by A1, A4, METRIC_1:2;
verum
end;
assume
x = y
; (dist_cart2S (X,Y)) . (x,y) = 0
then A5:
( (dist (x1,y1)) ^2 = 0 ^2 & (dist (x2,y2)) ^2 = 0 ^2 )
by METRIC_1:1;
(dist_cart2S (X,Y)) . (x,y) =
sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2))
by A1, Def10
.=
0
by A5, Lm1
;
hence
(dist_cart2S (X,Y)) . (x,y) = 0
; verum