let x, y, z be Element of [:REAL ,REAL :]; :: thesis: Eukl_dist2 . x,z <= (Eukl_dist2 . x,y) + (Eukl_dist2 . y,z)
reconsider x1 = x `1 , x2 = x `2 , y1 = y `1 , y2 = y `2 , z1 = z `1 , z2 = z `2 as Element of REAL by MCART_1:10;
A1: x = [x1,x2] by MCART_1:24;
set d5 = real_dist . x2,y2;
set d3 = real_dist . y1,z1;
set d1 = real_dist . x1,z1;
A2: y = [y1,y2] by MCART_1:24;
set d6 = real_dist . y2,z2;
set d4 = real_dist . x2,z2;
set d2 = real_dist . x1,y1;
A3: z = [z1,z2] by MCART_1:24;
real_dist . x2,z2 = abs (x2 - z2) by METRIC_1:def 13;
then 0 <= real_dist . x2,z2 by COMPLEX1:132;
then A4: (real_dist . x2,z2) ^2 <= ((real_dist . x2,y2) + (real_dist . y2,z2)) ^2 by METRIC_1:11, SQUARE_1:77;
( 0 <= (real_dist . x1,z1) ^2 & 0 <= (real_dist . x2,z2) ^2 ) by XREAL_1:65;
then A5: 0 + 0 <= ((real_dist . x1,z1) ^2 ) + ((real_dist . x2,z2) ^2 ) by XREAL_1:9;
real_dist . x1,z1 = abs (x1 - z1) by METRIC_1:def 13;
then 0 <= real_dist . x1,z1 by COMPLEX1:132;
then (real_dist . x1,z1) ^2 <= ((real_dist . x1,y1) + (real_dist . y1,z1)) ^2 by METRIC_1:11, SQUARE_1:77;
then ((real_dist . x1,z1) ^2 ) + ((real_dist . x2,z2) ^2 ) <= (((real_dist . x1,y1) + (real_dist . y1,z1)) ^2 ) + (((real_dist . x2,y2) + (real_dist . y2,z2)) ^2 ) by A4, XREAL_1:9;
then A6: sqrt (((real_dist . x1,z1) ^2 ) + ((real_dist . x2,z2) ^2 )) <= sqrt ((((real_dist . x1,y1) + (real_dist . y1,z1)) ^2 ) + (((real_dist . x2,y2) + (real_dist . y2,z2)) ^2 )) by A5, SQUARE_1:94;
real_dist . y2,z2 = abs (y2 - z2) by METRIC_1:def 13;
then A7: 0 <= real_dist . y2,z2 by COMPLEX1:132;
real_dist . x2,y2 = abs (x2 - y2) by METRIC_1:def 13;
then A8: 0 <= real_dist . x2,y2 by COMPLEX1:132;
real_dist . y1,z1 = abs (y1 - z1) by METRIC_1:def 13;
then A9: 0 <= real_dist . y1,z1 by COMPLEX1:132;
real_dist . x1,y1 = abs (x1 - y1) by METRIC_1:def 13;
then 0 <= real_dist . x1,y1 by COMPLEX1:132;
then sqrt ((((real_dist . x1,y1) + (real_dist . y1,z1)) ^2 ) + (((real_dist . x2,y2) + (real_dist . y2,z2)) ^2 )) <= (sqrt (((real_dist . x1,y1) ^2 ) + ((real_dist . x2,y2) ^2 ))) + (sqrt (((real_dist . y1,z1) ^2 ) + ((real_dist . y2,z2) ^2 ))) by A9, A8, A7, Th5;
then sqrt (((real_dist . x1,z1) ^2 ) + ((real_dist . x2,z2) ^2 )) <= (sqrt (((real_dist . x1,y1) ^2 ) + ((real_dist . x2,y2) ^2 ))) + (sqrt (((real_dist . y1,z1) ^2 ) + ((real_dist . y2,z2) ^2 ))) by A6, XXREAL_0:2;
then Eukl_dist2 . x,z <= (sqrt (((real_dist . x1,y1) ^2 ) + ((real_dist . x2,y2) ^2 ))) + (sqrt (((real_dist . y1,z1) ^2 ) + ((real_dist . y2,z2) ^2 ))) by A1, A3, Def9;
then Eukl_dist2 . x,z <= (Eukl_dist2 . x,y) + (sqrt (((real_dist . y1,z1) ^2 ) + ((real_dist . y2,z2) ^2 ))) by A1, A2, Def9;
hence Eukl_dist2 . x,z <= (Eukl_dist2 . x,y) + (Eukl_dist2 . y,z) by A2, A3, Def9; :: thesis: verum