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 by MCART_1:10;
A1: ( x = [x1,x2] & y = [y1,y2] ) by MCART_1:24;
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, Def9;
A3: ( 0 <= (real_dist . x1,y1) ^2 & 0 <= (real_dist . x2,y2) ^2 ) by XREAL_1:65;
then real_dist . x1,y1 = 0 by A2, Th2;
then A4: x1 = y1 by METRIC_1:9;
real_dist . x2,y2 = 0 by A2, A3, Th2;
hence x = y by A1, A4, METRIC_1:9; :: 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_1:9;
Eukl_dist2 . x,y = sqrt (((real_dist . x1,y1) ^2 ) + ((real_dist . x2,y2) ^2 )) by A1, Def9
.= 0 by A5, Th2 ;
hence Eukl_dist2 . x,y = 0 ; :: thesis: verum