let x, y, z be Element of [:REAL,REAL,REAL:]; :: thesis: Eukl_dist3 . (x,z) <= (Eukl_dist3 . (x,y)) + (Eukl_dist3 . (y,z))
reconsider x1 = x `1_3 , x2 = x `2_3 , x3 = x `3_3 , y1 = y `1_3 , y2 = y `2_3 , y3 = y `3_3 , z1 = z `1_3 , z2 = z `2_3 , z3 = z `3_3 as Element of REAL ;
A1: x = [x1,x2,x3] ;
set d9 = real_dist . (y3,z3);
set d5 = real_dist . (x2,y2);
set d6 = real_dist . (y2,z2);
set d1 = real_dist . (x1,z1);
set d2 = real_dist . (x1,y1);
A2: y = [y1,y2,y3] ;
real_dist . (y3,z3) = |.(y3 - z3).| by METRIC_1:def 12;
then A3: 0 <= real_dist . (y3,z3) by COMPLEX1:46;
real_dist . (y2,z2) = |.(y2 - z2).| by METRIC_1:def 12;
then A4: 0 <= real_dist . (y2,z2) by COMPLEX1:46;
real_dist . (x2,y2) = |.(x2 - y2).| by METRIC_1:def 12;
then A5: 0 <= real_dist . (x2,y2) by COMPLEX1:46;
set d7 = real_dist . (x3,z3);
set d8 = real_dist . (x3,y3);
set d3 = real_dist . (y1,z1);
set d4 = real_dist . (x2,z2);
A6: z = [z1,z2,z3] ;
real_dist . (x3,z3) = |.(x3 - z3).| by METRIC_1:def 12;
then 0 <= real_dist . (x3,z3) by COMPLEX1:46;
then A7: (real_dist . (x3,z3)) ^2 <= ((real_dist . (x3,y3)) + (real_dist . (y3,z3))) ^2 by METRIC_1:10, SQUARE_1:15;
real_dist . (x2,z2) = |.(x2 - z2).| by METRIC_1:def 12;
then 0 <= real_dist . (x2,z2) by COMPLEX1:46;
then A8: (real_dist . (x2,z2)) ^2 <= ((real_dist . (x2,y2)) + (real_dist . (y2,z2))) ^2 by METRIC_1:10, SQUARE_1:15;
real_dist . (x1,z1) = |.(x1 - z1).| by METRIC_1:def 12;
then 0 <= real_dist . (x1,z1) by COMPLEX1:46;
then (real_dist . (x1,z1)) ^2 <= ((real_dist . (x1,y1)) + (real_dist . (y1,z1))) ^2 by METRIC_1:10, SQUARE_1:15;
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 A8, XREAL_1:7;
then A9: (((real_dist . (x1,z1)) ^2) + ((real_dist . (x2,z2)) ^2)) + ((real_dist . (x3,z3)) ^2) <= ((((real_dist . (x1,y1)) + (real_dist . (y1,z1))) ^2) + (((real_dist . (x2,y2)) + (real_dist . (y2,z2))) ^2)) + (((real_dist . (x3,y3)) + (real_dist . (y3,z3))) ^2) by A7, XREAL_1:7;
( 0 <= (real_dist . (x1,z1)) ^2 & 0 <= (real_dist . (x2,z2)) ^2 ) by XREAL_1:63;
then A10: 0 + 0 <= ((real_dist . (x1,z1)) ^2) + ((real_dist . (x2,z2)) ^2) by XREAL_1:7;
0 <= (real_dist . (x3,z3)) ^2 by XREAL_1:63;
then 0 + 0 <= (((real_dist . (x1,z1)) ^2) + ((real_dist . (x2,z2)) ^2)) + ((real_dist . (x3,z3)) ^2) by A10, XREAL_1:7;
then A11: sqrt ((((real_dist . (x1,z1)) ^2) + ((real_dist . (x2,z2)) ^2)) + ((real_dist . (x3,z3)) ^2)) <= sqrt (((((real_dist . (x1,y1)) + (real_dist . (y1,z1))) ^2) + (((real_dist . (x2,y2)) + (real_dist . (y2,z2))) ^2)) + (((real_dist . (x3,y3)) + (real_dist . (y3,z3))) ^2)) by A9, SQUARE_1:26;
real_dist . (x3,y3) = |.(x3 - y3).| by METRIC_1:def 12;
then A12: 0 <= real_dist . (x3,y3) by COMPLEX1:46;
real_dist . (y1,z1) = |.(y1 - z1).| by METRIC_1:def 12;
then A13: 0 <= real_dist . (y1,z1) by COMPLEX1:46;
real_dist . (x1,y1) = |.(x1 - y1).| by METRIC_1:def 12;
then 0 <= real_dist . (x1,y1) by COMPLEX1:46;
then sqrt (((((real_dist . (x1,y1)) + (real_dist . (y1,z1))) ^2) + (((real_dist . (x2,y2)) + (real_dist . (y2,z2))) ^2)) + (((real_dist . (x3,y3)) + (real_dist . (y3,z3))) ^2)) <= (sqrt ((((real_dist . (x1,y1)) ^2) + ((real_dist . (x2,y2)) ^2)) + ((real_dist . (x3,y3)) ^2))) + (sqrt ((((real_dist . (y1,z1)) ^2) + ((real_dist . (y2,z2)) ^2)) + ((real_dist . (y3,z3)) ^2))) by A13, A5, A4, A12, A3, Lm2;
then sqrt ((((real_dist . (x1,z1)) ^2) + ((real_dist . (x2,z2)) ^2)) + ((real_dist . (x3,z3)) ^2)) <= (sqrt ((((real_dist . (x1,y1)) ^2) + ((real_dist . (x2,y2)) ^2)) + ((real_dist . (x3,y3)) ^2))) + (sqrt ((((real_dist . (y1,z1)) ^2) + ((real_dist . (y2,z2)) ^2)) + ((real_dist . (y3,z3)) ^2))) by A11, XXREAL_0:2;
then Eukl_dist3 . (x,z) <= (sqrt ((((real_dist . (x1,y1)) ^2) + ((real_dist . (x2,y2)) ^2)) + ((real_dist . (x3,y3)) ^2))) + (sqrt ((((real_dist . (y1,z1)) ^2) + ((real_dist . (y2,z2)) ^2)) + ((real_dist . (y3,z3)) ^2))) by A1, A6, Def22;
then Eukl_dist3 . (x,z) <= (Eukl_dist3 . (x,y)) + (sqrt ((((real_dist . (y1,z1)) ^2) + ((real_dist . (y2,z2)) ^2)) + ((real_dist . (y3,z3)) ^2))) by A1, A2, Def22;
hence Eukl_dist3 . (x,z) <= (Eukl_dist3 . (x,y)) + (Eukl_dist3 . (y,z)) by A2, A6, Def22; :: thesis: verum