reserve r, r1, r2, x, y, z,
        x1, x2, x3, y1, y2, y3 for Real;
reserve R, R1, R2, R3 for Element of 3-tuples_on REAL;
reserve p, q, p1, p2, p3, q1, q2 for Element of REAL 3;
reserve f1, f2, f3, g1, g2, g3, h1, h2, h3 for PartFunc of REAL,REAL;
reserve t, t0, t1, t2 for Real;

theorem
  |[ x1,x2,x3 ]| - |[ y1,y2,y3 ]| = (x1-y1)*<e1> + (x2-y2)*<e2> + (x3-y3)*<e3>
proof
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 thesis by A4,A5,Th36;
end;
