theorem Th30:
  r*p = r*p.1*<e1> + r*p.2*<e2> + r*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:  r*p.1*<e1> + r*p.2*<e2> + r*p.3*<e3>
     = |[ r*p.1*1,r*p.1*0,r*p.1*0 ]| + r*p.2*<e2> + r*p.3*<e3>
       by A1,Lm1
    .= |[ r*p.1,0,0 ]| + |[ r*p.2*0,r*p.2*1,r*p.2*0 ]| + r*p.3*<e3>
       by A2,Lm1
    .= |[ r*p.1,0,0 ]| + |[ 0,r*p.2,0 ]| + |[ r*p.3*0,r*p.3*0,r*p.3*1 ]|
       by A3,Lm1
    .= |[ r*p.1,0,0 ]| + |[ 0,r*p.2,0 ]| + |[ 0,0,r*p.3 ]|;
A5:  |[ r*p.1,0,0 ]|.1 = r*p.1 & |[ r*p.1,0,0 ]|.2 = 0 & |[ r*p.1,0,0 ]|.3 = 0;
A6:  |[ 0,r*p.2,0 ]|.1 = 0 & |[ 0,r*p.2,0 ]|.2 = r*p.2 & |[ 0,r*p.2,0 ]|.3 = 0;
A7:  |[ 0,0,r*p.3 ]|.1 = 0 & |[ 0,0,r*p.3 ]|.2 = 0 & |[ 0,0,r*p.3 ]|.3 = r*p.3;
A8:  r*p.1*<e1> + r*p.2*<e2> + r*p.3*<e3>
     = |[ r*p.1+0,0+r*p.2,0+0 ]| + |[ 0,0,r*p.3 ]| by A4,A5,A6,Lm2
    .= |[ r*p.1,r*p.2,0 ]| + |[ 0,0,r*p.3 ]|;
 |[ r*p.1,r*p.2,0 ]|.1 = r*p.1 & |[ r*p.1,r*p.2,0 ]|.2 = r*p.2 &
     |[ r*p.1,r*p.2,0 ]|.3 = 0;
     then r*p.1*<e1> + r*p.2*<e2> + r*p.3*<e3>
     = |[ r*p.1+0,r*p.2+0,0+r*p.3]| by A7,A8,Lm2
    .= |[ r*p.1,r*p.2,r*p.3 ]|;
     hence r*p = r*p.1*<e1> + r*p.2*<e2> + r*p.3*<e3> by Lm1;
end;
