let X, Y be non empty MetrSpace; :: thesis: 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:]; :: thesis: ( (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 ) :: thesis: ( x = y implies (dist_cart2S X,Y) . x,y = 0 )

proof

assume A2: (dist_cart2S X,Y) . x,y = 0 ; :: thesis: x = y
set d1 = dist x1,y1;
set d2 = dist x2,y2;
A3: sqrt (((dist x1,y1) ^2 ) + ((dist x2,y2) ^2 )) = 0 by A1, A2, Def1;
A4: 0 <= (dist x1,y1) ^2 by XREAL_1:65;
A5: 0 <= (dist x2,y2) ^2 by XREAL_1:65;
then A6: ( (dist x1,y1) ^2 = 0 & (dist x2,y2) ^2 = 0 ) by A3, A4, Th2;
(dist x1,y1) ^2 = 0 by A3, A4, A5, Th2;
then dist x1,y1 = 0 ;
then A7: x1 = y1 by METRIC_1:2;
dist x2,y2 = 0 by A6;
hence x = y by A1, A7, METRIC_1:2; :: thesis: verum

end;

assume A8: x = y ; :: thesis: (dist_cart2S X,Y) . x,y = 0
then A9: (dist x1,y1) ^2 = 0 ^2 by METRIC_1:1;
A10: (dist x2,y2) ^2 = 0 ^2 by A8, METRIC_1:1;
(dist_cart2S X,Y) . x,y = sqrt (((dist x1,y1) ^2 ) + ((dist x2,y2) ^2 )) by A1, Def1
.= 0 by A9, A10, Th2 ;
hence (dist_cart2S X,Y) . x,y = 0 ; :: thesis: verum