let X, Y, Z be non empty MetrSpace; for x, y being Element of [: the carrier of X, the carrier of Y, the carrier of Z:] holds
( (dist_cart3 (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:]; ( (dist_cart3 (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_cart3 (X,Y,Z)) . (x,y) = 0 implies x = y )
( x = y implies (dist_cart3 (X,Y,Z)) . (x,y) = 0 )proof
set d3 =
dist (
x3,
y3);
set d2 =
dist (
x2,
y2);
set d1 =
dist (
x1,
y1);
set d4 =
(dist (x1,y1)) + (dist (x2,y2));
assume
(dist_cart3 (X,Y,Z)) . (
x,
y)
= 0
;
x = y
then A2:
((dist (x1,y1)) + (dist (x2,y2))) + (dist (x3,y3)) = 0
by A1, Def4;
A3:
(
0 <= dist (
x1,
y1) &
0 <= dist (
x2,
y2) )
by METRIC_1:5;
then A4:
(
0 <= dist (
x3,
y3) &
0 + 0 <= (dist (x1,y1)) + (dist (x2,y2)) )
by METRIC_1:5, XREAL_1:7;
then A5:
(dist (x1,y1)) + (dist (x2,y2)) = 0
by A2, XREAL_1:27;
then
dist (
x1,
y1)
= 0
by A3, XREAL_1:27;
then A6:
x1 = y1
by METRIC_1:2;
dist (
x3,
y3)
= 0
by A2, A4, XREAL_1:27;
then A7:
x3 = y3
by METRIC_1:2;
dist (
x2,
y2)
= 0
by A3, A5, XREAL_1:27;
hence
x = y
by A1, A7, A6, METRIC_1:2;
verum
end;
assume A8:
x = y
; (dist_cart3 (X,Y,Z)) . (x,y) = 0
then A9:
( dist (x1,y1) = 0 & dist (x2,y2) = 0 )
by METRIC_1:1;
(dist_cart3 (X,Y,Z)) . (x,y) =
((dist (x1,y1)) + (dist (x2,y2))) + (dist (x3,y3))
by A1, Def4
.=
0
by A8, A9, METRIC_1:1
;
hence
(dist_cart3 (X,Y,Z)) . (x,y) = 0
; verum