reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);
reserve o,p,q,r,s,t for Point of TOP-REAL 3,
        M for Matrix of 3,F_Real;

theorem Th26:
  |{ a * p, q, r }| = a * |{ p, q, r }|
  proof
    |{ a * p,q , r }| = (a * p)`1 * q`2 * r`3 - (a * p)`3*q`2*r`1 -
                          (a * p)`1*q`3*r`2 + (a * p)`2*q`3*r`1 -
                          (a * p)`2*q`1*r`3 + (a * p)`3*q`1*r`2 by Th23
                     .= a * p`1 * q`2 * r`3 - (a * p)`3*q`2*r`1 -
                          (a * p)`1*q`3*r`2 + (a * p)`2*q`3*r`1 -
                          (a * p)`2*q`1*r`3 + (a * p)`3*q`1*r`2 by EUCLID_5:9
                     .= a * p`1 * q`2 * r`3 - a * p`3*q`2*r`1 -
                          (a * p)`1*q`3*r`2 + (a * p)`2*q`3*r`1 -
                          (a * p)`2*q`1*r`3 + (a * p)`3*q`1*r`2 by EUCLID_5:9
                     .= a * p`1 * q`2 * r`3 - a * p`3*q`2*r`1 -
                          a * p`1*q`3*r`2 + (a * p)`2*q`3*r`1 -
                          (a * p)`2*q`1*r`3 + (a * p)`3*q`1*r`2 by EUCLID_5:9
                     .= a * p`1 * q`2 * r`3 - a * p`3*q`2*r`1 -
                          a * p`1*q`3*r`2 + a * p`2*q`3*r`1 -
                          (a * p)`2*q`1*r`3 + (a * p)`3*q`1*r`2 by EUCLID_5:9
                     .= a * p`1 * q`2 * r`3 - a * p`3*q`2*r`1 -
                          a * p`1*q`3*r`2 + a * p`2*q`3*r`1 -
                          a * p`2*q`1*r`3 + (a * p)`3*q`1*r`2 by EUCLID_5:9
                      .= a * p`1 * q`2 * r`3 - a * p`3*q`2*r`1 -
                          a * p`1*q`3*r`2 + a * p`2*q`3*r`1 -
                          a * p`2*q`1*r`3 + a * p`3*q`1*r`2 by EUCLID_5:9
                      .= a * (p`1 * q`2 * r`3 - p`3*q`2*r`1 -
                          p`1*q`3*r`2 + p`2*q`3*r`1 - p`2*q`1*r`3 +
                          p`3*q`1*r`2)
                      .= a * |{ p,q,r }| by Th23;
    hence thesis;
  end;
