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
  p = p.1*<e1> + p.2*<e2> + p.3*<e3>
proof
A1:  <e1>.1 = 1 & <e1>.2 = 0 & <e1>.3 = 0;
A2:  <e2>.1 = 0 & <e2>.2 = 1 & <e2>.3 = 0;
A3:  <e3>.1 = 0 & <e3>.2 = 0 & <e3>.3 = 1;
A4:  p.1*<e1> + p.2*<e2> + p.3*<e3>
     = |[ p.1*1,p.1*0,p.1*0 ]| + p.2*<e2> + p.3*<e3> by A1,Lm1
    .= |[ p.1,0,0 ]| + |[ p.2*0,p.2*1,p.2*0 ]| + p.3*<e3> by A2,Lm1
    .= |[ p.1,0,0 ]| + |[ 0,p.2,0 ]| + |[ p.3*0,p.3*0,p.3*1 ]| by A3,Lm1
    .= |[ p.1,0,0 ]| + |[ 0,p.2,0 ]| + |[ 0,0,p.3 ]|;
A5:  |[ p.1,0,0 ]|.1 = p.1 & |[ p.1,0,0 ]|.2 = 0 & |[ p.1,0,0 ]|.3 = 0;
A6:  |[ 0,p.2,0 ]|.1 = 0 & |[ 0,p.2,0 ]|.2 = p.2 & |[ 0,p.2,0 ]|.3 = 0;
A7:  |[ 0,0,p.3 ]|.1 = 0 & |[ 0,0,p.3 ]|.2 = 0 & |[ 0,0,p.3 ]|.3 = p.3;
A8:  p.1*<e1> + p.2*<e2> + p.3*<e3>
     = |[ p.1+0,0+p.2,0+0 ]| + |[ 0,0,p.3 ]| by A4,A5,A6,Lm2
    .= |[ p.1,p.2,0 ]| + |[ 0,0,p.3 ]|;
 |[ p.1,p.2,0 ]|.1 = p.1 & |[ p.1,p.2,0 ]|.2 = p.2 &
     |[ p.1,p.2,0 ]|.3 = 0;
     then
p.1*<e1> + p.2*<e2> + p.3*<e3> = |[ p.1+0,p.2+0,0 +p.3]| by A7,A8,Lm2
    .= |[ p.1,p.2,p.3 ]|;
     hence p = p.1*<e1> + p.2*<e2> + p.3*<e3> by Th1;
end;
