let x1, x2, x3, y1, y2, y3 be Element of REAL ; :: thesis: |[x1,x2,x3]| + |[y1,y2,y3]| = |[(x1 + y1),(x2 + y2),(x3 + y3)]|
A1: |[y1,y2,y3]| . 1 = y1 by FINSEQ_1:45;
A2: |[y1,y2,y3]| . 2 = y2 by FINSEQ_1:45;
A3: |[y1,y2,y3]| . 3 = y3 by FINSEQ_1:45;
A4: (|[x1,x2,x3]| . 1) + (|[y1,y2,y3]| . 1) = x1 + y1 by A1, FINSEQ_1:45;
A5: (|[x1,x2,x3]| . 2) + (|[y1,y2,y3]| . 2) = x2 + y2 by A2, FINSEQ_1:45;
(|[x1,x2,x3]| . 3) + (|[y1,y2,y3]| . 3) = x3 + y3 by A3, FINSEQ_1:45;
hence |[x1,x2,x3]| + |[y1,y2,y3]| = |[(x1 + y1),(x2 + y2),(x3 + y3)]| by A4, A5, Lm2; :: thesis: verum