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

assume A2: Eukl_dist2 . x,y = 0 ; :: thesis: x = y
set d1 = real_dist . x1,y1;
set d2 = real_dist . x2,y2;
A3: sqrt (((real_dist . x1,y1) ^2 ) + ((real_dist . x2,y2) ^2 )) = 0 by A1, A2, Def9;
A4: 0 <= (real_dist . x1,y1) ^2 by XREAL_1:65;
A5: 0 <= (real_dist . x2,y2) ^2 by XREAL_1:65;
then A6: ( (real_dist . x1,y1) ^2 = 0 & (real_dist . x2,y2) ^2 = 0 ) by A3, A4, Th2;
(real_dist . x1,y1) ^2 = 0 by A3, A4, A5, Th2;
then real_dist . x1,y1 = 0 ;
then A7: x1 = y1 by METRIC_1:9;
real_dist . x2,y2 = 0 by A6;
hence x = y by A1, A7, METRIC_1:9; :: thesis: verum

end;

assume A8: x = y ; :: thesis: Eukl_dist2 . x,y = 0
then A9: (real_dist . x1,y1) ^2 = 0 ^2 by METRIC_1:9;
A10: (real_dist . x2,y2) ^2 = 0 ^2 by A8, METRIC_1:9;
Eukl_dist2 . x,y = sqrt (((real_dist . x1,y1) ^2 ) + ((real_dist . x2,y2) ^2 )) by A1, Def9
.= 0 by A9, A10, Th2 ;
hence Eukl_dist2 . x,y = 0 ; :: thesis: verum