let x, y be Element of [:REAL,REAL:]; :: thesis: ( 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 ) :: thesis: ( 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 ; :: thesis: 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; :: thesis: verum

end;

assume x = y ; :: thesis: 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 ;
hence Eukl_dist2 . (x,y) = 0 ; :: thesis: verum