let X, Y, Z be non empty MetrSpace; :: thesis: for x, y being Element of [: the carrier of X, the carrier of Y, the carrier of Z:] holds
( (dist_cart3S (X,Y,Z)) . (x,y) = 0 iff x = y )
let x, y be Element of [: the carrier of X, the carrier of Y, the carrier of Z:]; :: thesis: ( (dist_cart3S (X,Y,Z)) . (x,y) = 0 iff x = y )
reconsider x1 = x `1_3 , y1 = y `1_3 as Element of X ;
reconsider x2 = x `2_3 , y2 = y `2_3 as Element of Y ;
reconsider x3 = x `3_3 , y3 = y `3_3 as Element of Z ;
A1: ( x = [x1,x2,x3] & y = [y1,y2,y3] ) ;
thus ( (dist_cart3S (X,Y,Z)) . (x,y) = 0 implies x = y ) :: thesis: ( x = y implies (dist_cart3S (X,Y,Z)) . (x,y) = 0 )

proof

set d3 = dist (x3,y3);
set d2 = dist (x2,y2);
set d1 = dist (x1,y1);
A2: ( 0 <= (dist (x2,y2)) ^2 & 0 <= (dist (x3,y3)) ^2 ) by XREAL_1:63;
assume (dist_cart3S (X,Y,Z)) . (x,y) = 0 ; :: thesis: x = y
then sqrt ((((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2)) + ((dist (x3,y3)) ^2)) = 0 by A1, Def13;
then A3: sqrt (((dist (x1,y1)) ^2) + (((dist (x2,y2)) ^2) + ((dist (x3,y3)) ^2))) = 0 ;
( 0 <= (dist (x2,y2)) ^2 & 0 <= (dist (x3,y3)) ^2 ) by XREAL_1:63;
then A4: ( 0 <= (dist (x1,y1)) ^2 & 0 + 0 <= ((dist (x2,y2)) ^2) + ((dist (x3,y3)) ^2) ) by XREAL_1:7, XREAL_1:63;
then dist (x1,y1) = 0 by A3, Lm1;
then A5: x1 = y1 by METRIC_1:2;
A6: ((dist (x2,y2)) ^2) + ((dist (x3,y3)) ^2) = 0 by A3, A4, Lm1;
then dist (x2,y2) = 0 by A2, XREAL_1:27;
then A7: x2 = y2 by METRIC_1:2;
dist (x3,y3) = 0 by A6, A2, XREAL_1:27;
hence x = y by A1, A5, A7, METRIC_1:2; :: thesis: verum

end;

assume A8: x = y ; :: thesis: (dist_cart3S (X,Y,Z)) . (x,y) = 0
then A9: ( (dist (x1,y1)) ^2 = 0 ^2 & (dist (x2,y2)) ^2 = 0 ^2 ) by METRIC_1:1;
(dist_cart3S (X,Y,Z)) . (x,y) = sqrt ((((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2)) + ((dist (x3,y3)) ^2)) by A1, Def13
.= 0 ^2 by A8, A9, METRIC_1:1, SQUARE_1:17 ;
hence (dist_cart3S (X,Y,Z)) . (x,y) = 0 ; :: thesis: verum