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 by MCART_1:10;
reconsider x2 = x `2 , y2 = y `2 as Element of Y by MCART_1:10;
A1:
( x = [x1,x2] & y = [y1,y2] )
by MCART_1:24;
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, Def1;
A3:
(
0 <= (dist x1,y1) ^2 &
0 <= (dist x2,y2) ^2 )
by XREAL_1:65;
then
dist x1,
y1 = 0
by A2, Th2;
then A4:
x1 = y1
by METRIC_1:2;
dist x2,
y2 = 0
by A2, A3, Th2;
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, Def1
.=
0
by A5, Th2
;
hence
(dist_cart2S X,Y) . x,y = 0
; verum