let x1, x2, y1, y2, z1, z2 be Real; :: thesis: |[x1,y1,z1]| + |[x2,y2,z2]| = |[(x1 + x2),(y1 + y2),(z1 + z2)]|
A1: |[x2,y2,z2]| . 1 = x2 by FINSEQ_1:45;
A2: |[x2,y2,z2]| . 2 = y2 by FINSEQ_1:45;
A3: |[x2,y2,z2]| . 3 = z2 by FINSEQ_1:45;
A4: (|[x1,y1,z1]| . 1) + (|[x2,y2,z2]| . 1) = x1 + x2 by A1, FINSEQ_1:45;
A5: (|[x1,y1,z1]| . 2) + (|[x2,y2,z2]| . 2) = y1 + y2 by A2, FINSEQ_1:45;
(|[x1,y1,z1]| . 3) + (|[x2,y2,z2]| . 3) = z1 + z2 by A3, FINSEQ_1:45;
hence |[x1,y1,z1]| + |[x2,y2,z2]| = |[(x1 + x2),(y1 + y2),(z1 + z2)]| by A4, A5, EUCLID_8:55; :: thesis: verum