let x1, x2, x3, y1, y2, y3 be Real; :: thesis: |[x1,x2,x3]| - |[y1,y2,y3]| = (((x1 - y1) * <e1>) + ((x2 - y2) * <e2>)) + ((x3 - y3) * <e3>)
A4: (|[x1,x2,x3]| . 1) - (|[y1,y2,y3]| . 1) = x1 - y1 ;
A5: (|[x1,x2,x3]| . 2) - (|[y1,y2,y3]| . 2) = x2 - y2 ;
(|[x1,x2,x3]| . 3) - (|[y1,y2,y3]| . 3) = x3 - y3 ;
hence |[x1,x2,x3]| - |[y1,y2,y3]| = (((x1 - y1) * <e1>) + ((x2 - y2) * <e2>)) + ((x3 - y3) * <e3>) by A4, A5, Th36; :: thesis: verum