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 Th32:
  r*|[ x,y,z ]| = r*x*<e1> + r*y*<e2> + r*z*<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;
    set p = |[ x,y,z ]|;
 A4:  r*x*<e1> + r*y*<e2> + r*z*<e3>
      = |[ r*x*1,r*x*0,r*x*0 ]| + r*y*<e2> + r*z*<e3> by A1,Lm1
     .= |[ r*x,0,0 ]| + |[ r*y*0,r*y*1,r*y*0 ]| + r*z*<e3> by A2,Lm1
     .= |[ r*x,0,0 ]| + |[ 0,r*y,0 ]| + |[ r*z*0,r*z*0,r*z*1 ]| by A3,Lm1
     .= |[ r*x,0,0 ]| + |[ 0,r*y,0 ]| + |[ 0,0,r*z ]|;
 A5:  |[ r*x,0,0 ]|.1 = r*x & |[ r*x,0,0 ]|.2 = 0 & |[ r*x,0,0 ]|.3 = 0;
 A6:  |[ 0,r*y,0 ]|.1 = 0 & |[ 0,r*y,0 ]|.2 = r*y & |[ 0,r*y,0 ]|.3 = 0;
 A7:  |[ 0,0,r*z ]|.1 = 0 & |[ 0,0,r*z ]|.2 = 0 & |[ 0,0,r*z ]|.3 = r*z;
 A8: r*x*<e1> + r*y*<e2> + r*z*<e3>
      = |[ r*x+0,r*y+0,0+0 ]| + |[ 0,0,r*z ]| by A4,A5,A6,Lm2
     .= |[ r*x,r*y,0 ]| + |[ 0,0,r*z ]|;
  |[ r*x,r*y,0 ]|.1 = r*x & |[ r*x,r*y,0 ]|.2 = r*y & |[ r*x,r*y,0 ]|.3 = 0;
      then r*x*<e1> + r*y*<e2> + r*z*<e3>
       = |[ r*x+0,r*y+0,0+r*z ]| by A7,A8,Lm2
      .= |[ r*x,r*y,r*z ]|;
      hence r*p = r*x*<e1> + r*y*<e2> + r*z*<e3> by Lm6;
end;
