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 Th31:
  |[ x,y,z ]| = x*<e1> + y*<e2> + 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:  x*<e1> + y*<e2> + z*<e3> = |[ x*1,x*0,x*0 ]| + y*<e2> + z*<e3> by A1,Lm1
     .= |[ x,0,0 ]| + |[ y*0,y*1,y*0 ]| + z*<e3> by A2,Lm1
     .= |[ x,0,0 ]| + |[ 0,y,0 ]| + |[ z*0,z*0,z*1 ]| by A3,Lm1
     .= |[ x,0,0 ]| + |[ 0,y,0 ]| + |[ 0,0,z ]|;
 A5:  |[ x,0,0 ]|.1 = x & |[ x,0,0 ]|.2 = 0 & |[ x,0,0 ]|.3 = 0;
 A6:  |[ 0,y,0 ]|.1 = 0 & |[ 0,y,0 ]|.2 = y & |[ 0,y,0 ]|.3 = 0;
 A7:  |[ 0,0,z ]|.1 = 0 & |[ 0,0,z ]|.2 = 0 & |[ 0,0,z ]|.3 = z;
 A8: x*<e1> + y*<e2> + z*<e3>
      = |[ x+0,y+0,0+0 ]| + |[ 0,0,z ]| by A4,A5,A6,Lm2
     .= |[ x,y,0 ]| + |[ 0,0,z ]|;
  |[ x,y,0 ]|.1 = x & |[ x,y,0 ]|.2 = y & |[ x,y,0 ]|.3 = 0;
      then x*<e1> + y*<e2> + z*<e3> = |[ x+0,y+0,0+z ]| by A7,A8,Lm2
       .= |[ x,y,z ]|;
      hence p = x*<e1> + y*<e2> + z*<e3>;
end;
