let X, Y, Z, W be non empty MetrSpace; for x, y being Element of [:the carrier of X,the carrier of Y,the carrier of Z,the carrier of W:] holds
( (dist_cart4 X,Y,Z,W) . x,y = 0 iff x = y )
let x, y be Element of [:the carrier of X,the carrier of Y,the carrier of Z,the carrier of W:]; ( (dist_cart4 X,Y,Z,W) . 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 ;
reconsider x3 = x `3 , y3 = y `3 as Element of Z ;
reconsider x4 = x `4 , y4 = y `4 as Element of W ;
A1:
( x = [x1,x2,x3,x4] & y = [y1,y2,y3,y4] )
by MCART_1:60;
thus
( (dist_cart4 X,Y,Z,W) . x,y = 0 implies x = y )
( x = y implies (dist_cart4 X,Y,Z,W) . x,y = 0 )proof
set d1 =
dist x1,
y1;
set d2 =
dist x2,
y2;
set d3 =
dist x3,
y3;
set d5 =
dist x4,
y4;
set d4 =
(dist x1,y1) + (dist x2,y2);
set d6 =
(dist x3,y3) + (dist x4,y4);
A2:
(
0 <= dist x3,
y3 &
0 <= dist x4,
y4 )
by METRIC_1:5;
then A3:
0 + 0 <= (dist x3,y3) + (dist x4,y4)
by XREAL_1:9;
assume
(dist_cart4 X,Y,Z,W) . x,
y = 0
;
x = y
then A4:
((dist x1,y1) + (dist x2,y2)) + ((dist x3,y3) + (dist x4,y4)) = 0
by A1, Def7;
A5:
(
0 <= dist x1,
y1 &
0 <= dist x2,
y2 )
by METRIC_1:5;
then A6:
0 + 0 <= (dist x1,y1) + (dist x2,y2)
by XREAL_1:9;
then A7:
(dist x1,y1) + (dist x2,y2) = 0
by A4, A3, XREAL_1:29;
then
dist x2,
y2 = 0
by A5, XREAL_1:29;
then A8:
x2 = y2
by METRIC_1:2;
A9:
(dist x3,y3) + (dist x4,y4) = 0
by A4, A6, A3, XREAL_1:29;
then
dist x3,
y3 = 0
by A2, XREAL_1:29;
then A10:
x3 = y3
by METRIC_1:2;
dist x4,
y4 = 0
by A2, A9, XREAL_1:29;
then A11:
x4 = y4
by METRIC_1:2;
dist x1,
y1 = 0
by A5, A7, XREAL_1:29;
hence
x = y
by A1, A8, A10, A11, METRIC_1:2;
verum
end;
assume A12:
x = y
; (dist_cart4 X,Y,Z,W) . x,y = 0
then A13:
( dist x2,y2 = 0 & dist x3,y3 = 0 )
by METRIC_1:1;
A14:
dist x4,y4 = 0
by A12, METRIC_1:1;
(dist_cart4 X,Y,Z,W) . x,y =
((dist x1,y1) + (dist x2,y2)) + ((dist x3,y3) + (dist x4,y4))
by A1, Def7
.=
0
by A12, A13, A14, METRIC_1:1
;
hence
(dist_cart4 X,Y,Z,W) . x,y = 0
; verum