let x, y be Element of [:REAL,REAL:]; ( Eukl_dist2 . (x,y) = 0 iff x = y )
reconsider x1 = x `1 , x2 = x `2 , y1 = y `1 , y2 = y `2 as Element of REAL ;
A1:
( x = [x1,x2] & y = [y1,y2] )
;
thus
( Eukl_dist2 . (x,y) = 0 implies x = y )
( x = y implies Eukl_dist2 . (x,y) = 0 )proof
set d2 =
real_dist . (
x2,
y2);
set d1 =
real_dist . (
x1,
y1);
assume
Eukl_dist2 . (
x,
y)
= 0
;
x = y
then A2:
sqrt (((real_dist . (x1,y1)) ^2) + ((real_dist . (x2,y2)) ^2)) = 0
by A1, Def18;
A3:
(
0 <= (real_dist . (x1,y1)) ^2 &
0 <= (real_dist . (x2,y2)) ^2 )
by XREAL_1:63;
then
real_dist . (
x1,
y1)
= 0
by A2, Lm1;
then A4:
x1 = y1
by METRIC_1:8;
real_dist . (
x2,
y2)
= 0
by A2, A3, Lm1;
hence
x = y
by A1, A4, METRIC_1:8;
verum
end;
assume
x = y
; Eukl_dist2 . (x,y) = 0
then A5:
( (real_dist . (x1,y1)) ^2 = 0 ^2 & (real_dist . (x2,y2)) ^2 = 0 ^2 )
by METRIC_1:8;
Eukl_dist2 . (x,y) =
sqrt (((real_dist . (x1,y1)) ^2) + ((real_dist . (x2,y2)) ^2))
by A1, Def18
.=
0
by A5, Lm1
;
hence
Eukl_dist2 . (x,y) = 0
; verum