reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;

theorem Th19:
  a * |(u,v)| = |(a * u,v)| & a * |(u,v)| = |(u, a * v)|
  proof
A1: a * u`1 = a * u.1 by EUCLID_5:def 1
           .= (a * u).1 by RVSUM_1:44
           .= (a * u)`1 by EUCLID_5:def 1;
A2: a * u`2 = a * u.2 by EUCLID_5:def 2
           .= (a * u).2 by RVSUM_1:44
           .= (a * u)`2 by EUCLID_5:def 2;
A3: a * u`3 = a * u.3 by EUCLID_5:def 3
           .= (a * u).3 by RVSUM_1:44
           .= (a * u)`3 by EUCLID_5:def 3;
    thus a * |(u,v)| = a * (u`1*v`1 + u`2*v`2+u`3*v`3) by EUCLID_5:29
                    .= (a *u)`1 * v`1 + (a*u)`2 * v`2 + (a*u)`3 * v`3
                       by A1,A2,A3
                    .= |(a * u,v)| by EUCLID_5:29;
A4: a * v`1 = a * v.1 by EUCLID_5:def 1
           .= (a * v).1 by RVSUM_1:44
           .= (a * v)`1 by EUCLID_5:def 1;
A5: a * v`2 = a * v.2 by EUCLID_5:def 2
           .= (a * v).2 by RVSUM_1:44
           .= (a * v)`2 by EUCLID_5:def 2;
A6: a * v`3 = a * v.3 by EUCLID_5:def 3
           .= (a * v).3 by RVSUM_1:44
           .= (a * v)`3 by EUCLID_5:def 3;
    thus a * |(u,v)| = a * (u`1*v`1 + u`2*v`2+u`3*v`3) by EUCLID_5:29
                    .= u`1 * (a * v)`1 + u`2 * (a * v)`2 + u`3 * (a * v)`3
                      by A4,A5,A6
                    .= |(u,a * v)| by EUCLID_5:29;
  end;
